## Course Standards

## General Course Information and Notes

### General Notes

Intensive courses have been designed so that the teacher will select the appropriate standards when developing curricula tailored to meet the needs of individual students, taking into account their grade and instructional level. This course should not be used in place of a core mathematics course but is intended to provide intervention for students who require extra mathematics instruction.

**English Language Development ELD Standards Special Notes Section:**

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

http://www.cpalms.org/uploads/docs/standards/eld/MA.pdf

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

**Additional Instructional Resources:**

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

### General Information

**Course Number:**1204000

**Course Path:**

**Abbreviated Title:**M/J INTENS MATH (MC)

**Course Length:**Year (Y)

**Course Attributes:**

- Class Size Core Required

**Course Status:**Course Approved

**Grade Level(s):**6,7,8

## Educator Certifications

## Student Resources

## Original Student Tutorials

Join in on creating a delicious sundae by adding decimals to the thousandths, using the standard algorithm, in this interactive tutorial.

Type: Original Student Tutorial

Learn how equations can have 1 solution, no solution or infinitely many solutions in this interactive tutorial.

This is part five of five in a series on solving multi-step equations.

- Click
**HERE**to open Part 1: Combining Like Terms - Click
**HERE**to open Part 2: The Distributive Property - Click
**HERE**to open Part 3: Variables on Both Sides - Click
**HERE**to open Part 4: Putting It All Together - [CURRENT TUTORIAL] Part 5: How Many Solutions?

Type: Original Student Tutorial

Learn alternative methods of solving multi-step equations in this interactive tutorial.

This is part five of five in a series on solving multi-step equations.

- Click
**HERE**to open Part 1: Combining Like Terms - Click
**HERE**to open Part 2: The Distributive Property - Click
**HERE**to open Part 3: Variables on Both Sides - [CURRENT TUTORIAL] Part 4: Putting It All Together
- Click
**HERE**to open Part 5: How Many Solutions?

Type: Original Student Tutorial

Learn how to calculate the volume of spheres while learning how they make Bubble Tea in this interactive tutorial.

Type: Original Student Tutorial

Learn how to solve multi-step equations that contain variables on both sides of the equation in this interactive tutorial.

This is part five of five in a series on solving multi-step equations.

- Click
**HERE**to open Part 1: Combining Like Terms - Click
**HERE**to open Part 2: The Distributive Property - [CURRENT TUTORIAL] Part 3: Variables on Both Sides
- Click
**HERE**to open Part 4: Putting It All Together - Click
**HERE**to open Part 5: How Many Solutions?

Type: Original Student Tutorial

Explore how to solve multi-step equations using the distributive property in this interactive tutorial.

This is part five of five in a series on solving multi-step equations.

- Click
**HERE**to open Part 1: Combining Like Terms - [CURRENT TUTORIAL] Part 2: The Distributive Property
- Click
**HERE**to open Part 3: Variables on Both Sides - Click
**HERE**to open Part 4: Putting It All Together - Click
**HERE**to open Part 5: How Many Solutions?

Type: Original Student Tutorial

Cruise along as you discover how to qualitatively describe functions in this interactive tutorial.

Type: Original Student Tutorial

Learn how to solve multi-step equations that contain like terms in this interactive tutorial.

This is part one of five in a series on solving multi-step equations.

- [CURRENT TUTORIAL] Part 1: Combining Like Terms
- Click
**HERE**to open Part 2: The Distributive Property - Click
**HERE**to open Part 3: Variables on Both Sides - Click
**HERE**to open Part 4: Putting It All Together - Click
**HERE**to open Part 5: How Many Solutions?

Type: Original Student Tutorial

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 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

Learn how to solve 1-step multiplication and division equations with Dr. E. Quation in Part 2 of this series of interactive tutorials. You'll also learn how to check your answers to make sure your answer is the solution to the equation.

Type: Original Student Tutorial

See how sweet it can be to compare two functions in this interactive tutorial. Compare linear functions by looking at verbal descriptions, tables of values, equations and graphical forms to see which function has a greater rate of change.

Type: Original Student Tutorial

Learn how to solve and check one-step addition and subtraction equations with Dr. E. Quation as you complete this interactive tutorial.

**Click here to open Dr. E. Quation Part 2: One-Step Multiplication and Division Equations**

Type: Original Student Tutorial

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

Type: Original Student Tutorial

Have some fun with FUNctions! Learn how to identify linear and non-linear functions in this interactive tutorial.

Type: Original Student Tutorial

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

Learn how to interpret histograms to analyze data, and help an inventor predict the range of a catapult in part 2 of this interactive tutorial series. More specifically, you'll learn to describe the shape and spread of data distributions.

Click **HERE** to open part 1.

Type: Original Student Tutorial

Learn how to create a histogram to display continuous data from projectiles launched by a catapult in this interactive tutorial.

This is part 1 in a 2-part series. Click **HERE** to open part 2.

Type: Original Student Tutorial

Explore computer coding on the farm by using IF statements and repeat loops to evaluate mathematical expressions. In this interactive tutorial, you'll also solve problems involving inequalities.

Click below to check out the other tutorials in the series.

Type: Original Student Tutorial

Explore computer coding on the farm by using relational operators and IF statements to evaluate expressions. In this interactive tutorial, you'll also solve problems involving inequalities.

Click below to check out the other tutorials in the series.

Type: Original Student Tutorial

Explore computer coding on the farm by using condition and IF statements in this interactive tutorial. You'll also get a chance to apply the order of operations as you using coding to solve problems.

Click below to check out the other tutorials in the series.

Type: Original Student Tutorial

Explore computer coding on the farm by declaring and initializing variables in this interactive tutorial. You'll also get a chance to practice your long division skills.

Type: Original Student Tutorial

Learn how to calculate and interpret the Mean Absolute Deviation (MAD) of data sets in this travel-themed, interactive statistics tutorial.

Type: Original Student Tutorial

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

Type: Original Student Tutorial

Describe the average velocity of a dune buggy using kinematics in this interactive tutorial. You'll calculate displacement and average velocity, create and analyze a velocity vs. time scatterplot, and relate average velocity to the slope of position vs. time scatterplots.

This is part 3 of 3 in a series that mirrors inquiry-based, hands-on activities from our popular workshops.

- Click HERE to open The Notion of Motion, Part 1 - Time Measurements
- Click HERE to open The Notion of Motion, Part 2 - Position vs Time

Type: Original Student Tutorial

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

Learn how to calculate the probability of simple events, that probability is the likeliness of an event occurring and that some events may be more likely than others to occur, in this interactive tutorial.

Type: Original Student Tutorial

Compare multiple samples of lionfish to make generalizations about the population by analyzing the samples’ Mean Absolute Deviations and their distributions in this interactive tutorial.

Type: Original Student Tutorial

Continue an exploration of kinematics to describe linear motion by focusing on position-time measurements from the motion trial in part 1. You'll identify position measurements from the spark tape, analyze a scatterplot of the position-time data, calculate and interpret slope on the position-time graph, and make inferences about the dune buggy’s average speed

Type: Original Student Tutorial

Help Alice discover that compound probabilities can be determined through calculations or by drawing tree diagrams in this interactive tutorial.

Type: Original Student Tutorial

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

Learn how to use the equation of a linear trend line to interpolate and extrapolate bivariate data plotted in a scatterplot. You will see the usefulness of trend lines and how they are used in this online tutorial.

Type: Original Student Tutorial

Explore how to interpret the slope and y-intercept of a linear trend line when bivariate data is graphed on a scatterplot in this interactive tutorial.

Type: Original Student Tutorial

Learn how to write the equation of a linear trend line when fitted to bivariate data in a scatterplot in this online tutorial.

Type: Original Student Tutorial

Explore informally fitting a trend line to data graphed in a scatter plot in this interactive online tutorial.

This is part 3 in 6-part series. Click below to open the other tutorials in the series.

- Scatterplots Part 1: Graphing
- Scatterplots Part 2
- Scatterplots Part 3: Trend Lines
- Scatterolots Part 4: Equation of the Trend Line
- Scatterplots Part 5: Interpreting the Equation of the Trend Line
- Scatterplots Part 6: Using Linear Models

Type: Original Student Tutorial

Explore the different types of associations that can exist between bivariate data in this interactive tutorial.

Type: Original Student Tutorial

Learn how to graph bivariate data in a scatterplot in this interactive tutorial.

Type: Original Student Tutorial

Learn how to make and interpret boxplots in this pet-themed, interactive tutorial.

Type: Original Student Tutorial

Learn how writers and speakers create arguments by stating a claim and backing it up with reasons and evidence. In this interactive tutorial, you'll hear speeches from candidates for Student Council President and complete practice exercises.

Type: Original Student Tutorial

Learn how to use probability to predict expected outcomes at the Carnival in this interactive tutorial.

Type: Original Student Tutorial

Discover how to calculate and interpret the mean, median, mode and range of data sets from the zoo in this interactive tutorial.

Type: Original Student Tutorial

Learn to describe a sequence of transformations that will produce similar figures. This interactive tutorial will allow you to practice with rotations, translations, reflections, and dilations.

Type: Original Student Tutorial

Learn how to identify explicit evidence and understand implicit meaning in a text.

You will be able to organize information in a table and write ratios equivalent to a given ratio in order to solve real-world and mathematical problems.

Type: Original Student Tutorial

Learn how to evaluate the *soundness* of several speakers' arguments as they debate whether or not the driving age should be raised from 16 years old to 18 or even higher with this interactive tutorial.

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

Investigate how temperature affects the rate of chemical reactions in this interactive tutorial.

Type: Original Student Tutorial

Learn what genetic engineering is and some of the applications of this technology. In this interactive tutorial, you’ll gain an understanding of some of the benefits and potential drawbacks of genetic engineering. Ultimately, you’ll be able to think critically about genetic engineering and write an argument describing your own perspective on its impacts.

Type: Original Student Tutorial

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

Type: Original Student Tutorial

Learn how to 1) create and use number lines with positive and negative numbers, 2) use those number lines to graph positive and negative numbers, 3) find their distance from zero, 4) find a number’s opposite using a number line and using number signs, and 5) recognize that zero is its own opposite.

Type: Original Student Tutorial

Learn to construct linear functions from tables that contain sets of data that relate to each other in special ways as you complete this interactive tutorial.

Type: Original Student Tutorial

Howdy y’all! I’m Deputy Design, a cowboy architect. I am going to use my architectural scale drawings for a new horse arena to teach you how to solve problems involving scale drawings. In this tutorial, you will learn to calculate actual lengths using a scale and proportions.

Type: Original Student Tutorial

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

Learn how to identify explicit evidence and understand implicit meaning in a text.

In this tutorial, you will learn how to identify a speaker’s argument or claim. You will also learn how to evaluate the evidence and reasoning presented in a speech.

Type: Original Student Tutorial

## Educational Games

In this challenge game, you will be solving equations with variables on both sides. Each equation has a real solution. Use the "Teach Me" button to review content before the challenge. 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

Challenge yourself with this Prodigi game to see if you can answer questions about points and lines in graphs. Practice using slope-intercept form of a line. Try the "Teach Me" button to prepare yourself. When you are ready, play Prodigi! Be sure to use the review function at the end for any incorrect answers! Have fun!

Type: Educational Game

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

Challenge yourself with this Prodigi game to see if you can answer questions about experimental probability. Try the "Teach Me" button to prepare yourself. When you are ready, play Prodigi! You get one free solve and two hints. Be sure to use the review function at the end for the solution to any incorrect answer! Have fun!

Type: Educational Game

Challenge yourself with this Prodigi game to see if you can answer questions about probability of single events. Try the "Teach Me" button to prepare yourself. When you are ready, play Prodigi! You get one free solve and two hints. Be sure to use the review function at the end for the solution to any incorrect answer! Have fun!

Type: Educational Game

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

Type: Educational Game

Play this robot game to improve your negative number skills. Order and compare negative and positive numbers to complete missions in order to weaken the robots who have taken over the world.

Type: Educational Game

This fun and interactive game helps practice estimation skills, using various operations of choice, including addition, subtraction, multiplication, division, using decimals, fractions, and percents.

Various levels of difficulty make this game appropriate for multiple age and ability levels.

*Addition/**Subtraction:* The addition and subtraction of whole numbers, the addition and subtraction of decimals.

*Multiplication/Division: *The multiplication and addition of fractions and decimals.

*Percentages: *Identify the percentage of a whole number.

*Fractions: *Multiply and divde a whole number by a fraction, as well as apply properties of operations.

Type: Educational Game

This is a fun and interactive game that helps students practice ordering rational numbers, including decimals, fractions, and percents. You are planting and harvesting flowers for cash. Allow the bee to pollinate, and you can multiply your crops and cash rewards!

Type: Educational Game

This addition game uses mixed decimals to the tenths place. This game encourages some logical analysis as well as addition skills. There may be several ways to make the first couple of circles sum to 3, but there is only one way to combine all the given numbers so that every circle sums to 3.

Type: Educational Game

Test your fraction skills by answering questions on this site. This quiz asks you to simplify fractions, convert fractions to decimals and percentages, and answer algebra questions involving fractions. You can even choose difficulty level, question types, and time limit.

Type: Educational 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

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

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

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

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 poses problems requiring the students to position numbers in a diagram, so all numbers in a line add up to a given value.

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

In this activity, students enter coordinates to make a path to get to a target destination while avoiding mines. This activity allows students to explore Cartesian coordinates and the Cartesian coordinate plane. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Educational Game

## Educational Software / Tools

This 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

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

This resource is an online glossary to find the meaning of math terms. Students can also use the online glossary to find words that are related to the word typed in the search box. For example: Type in "transversal" and 11 other terms will come up. Click on one of those terms and its meaning is displayed.

Type: Educational Software / Tool

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

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

Type: Perspectives Video: Expert

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

Type: Perspectives Video: Expert

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

Type: Perspectives Video: Expert

It's impossible to count every animal in a park, but with statistics and some engineering, biologists can come up with a good estimate.

Type: Perspectives Video: Expert

How do scientists collect information from the world? They sample it! Learn how scientists take samples of phytoplankton not only to monitor their populations, but also to make inferences about the rest of the ecosystem!

Type: Perspectives Video: Expert

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

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

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

Type: Perspectives Video: Professional/Enthusiast

Find out how math and technology can help you (try to) get away from civilization.

Type: Perspectives Video: Professional/Enthusiast

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

Type: Perspectives Video: Professional/Enthusiast

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

Sometimes scientists conduct a census, too! Learn how population sampling can help monitor the progress of an ecological restoration project.

Type: Perspectives Video: Professional/Enthusiast

## Presentation/Slideshow

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

## Problem-Solving Tasks

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.

Type: Problem-Solving Task

In this task students are given a tile pattern involving congruent regular octagons and squares. They are asked to determine the interior angle measure of the octagon and verify the attributes of the square.

Type: Problem-Solving Task

Students are asked to use a diagram or table to write an algebraic expression and use the expression to solve problems.

Type: Problem-Solving Task

This task asks students to calculate probabilities using information presented in a two-way frequency table.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

This tasks gives a verbal description for computing the perimeter of a rectangle and asks the students to find an expression for this perimeter. They then have to use the expression to evaluate the perimeter for specific values of the two variables.

Type: Problem-Solving Task

Students are asked to determine if given expressions are equivalent.

Type: Problem-Solving Task

The purpose of this task is to ask students to write expressions and to consider what it means for two expressions to be equivalent.

Type: Problem-Solving Task

Students are asked to explore and then write an expression with an exponent. The purpose of this task is to introduce the idea of exponential growth and then connect that growth to expressions involving exponents. It illustrates well how fast exponential expressions grow.

Type: Problem-Solving Task

This problem asks the student to find a 3% sales tax on a vase valued at $450.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Students are asked to determine and illustrate all possible descriptions for the base and height of a given triangle.

Type: Problem-Solving Task

Students are asked to draw a scale model of a building and find related volume and surface areas of the model and the building which are rectangular prisms.

Type: Problem-Solving Task

Students are asked to demonstrate two different strategies for finding the area of polygons shown on grids.

Type: Problem-Solving Task

Students are asked to use the given information to determine the cost of painting a barn.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

The purpose of this task is to gain a better understanding of factors and common factors. Students should use the distributive property to show that the sum of two numbers that have a common factor is also a multiple of the common factor.

Type: Problem-Solving Task

Students are asked to reason about and explain the position of two locations relative to sea level.

Type: Problem-Solving Task

The purpose of this task is for students to solve problems involving decimals in a context involving a concept that supports financial literacy, namely inflation. Inflation is a sustained increase in the average price level. In this task, students are asked to compare the buying power of $20 in 1987 and 2012, at least with respect to movie tickets.

Type: Problem-Solving Task

Given the fact 13 x 17 = 221, students are asked to reason about and explain the decimal placement in multiplication and division problems where some of the numbers involved have been changed by powers of ten.

Type: Problem-Solving Task

Students are asked to reason about and explain the placement of decimals in quotients.

Type: Problem-Solving Task

Students are asked to solve a distance problem involving fractions.

Type: Problem-Solving Task

Students are asked to solve a fraction division problem using both a visual model and the standard algorithm within a real-world context.

Type: Problem-Solving Task

The purpose of this task is converting square units. Use the information provided to answer the questions posed. Since this task asks students to critique Jada's reasoning, it provides an opportunity to work on Standard for Mathematical Practice MAFS.K12.MP.3.1 - Construct Viable Arguments and Critique the Reasoning of Others.

Type: Problem-Solving Task

Students are asked to use a ratio to determine how much money Jim and Jesse had at the start of their trip.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Students are asked to determine the percent of the area of a store covered by a security camera. Then, students are asked to determine the "best" place to position the camera and support their answer.

Type: Problem-Solving Task

Use the information provided to find out the original price of Selina's shirt. There are several different ways to reason through this problem; two approaches are shown.

Type: Problem-Solving Task

This problem is the fifth in a series of seven about ratios. At first glance the problem may look to be beyond MAFS.6.RP.1.3, which limits itself to "describe a ratio relationship between two quantities." However, even though there are three quantities (the number of each candidates' votes), they are only considered two at a time.

Type: Problem-Solving Task

This is the sixth problem in a series of seven that use the context of a classroom election. While it still deals with simple ratios and easily managed numbers, the mathematics surrounding the ratios are increasingly complex. In this problem, the students are asked to determine the difference in votes received by two of the three candidates.

Type: Problem-Solving Task

This is the last problem of seven in a series about ratios set in the context of a classroom election. Since the number of voters is not known, the problem is quite abstract and requires a deep understanding of ratios and their relationship to fractions.

Type: Problem-Solving Task

This problem is the third in a series of tasks set in the context of a class election. Students are given a ratio and total number of voters and are asked to determine the difference between the winning number of votes received and the number of votes needed for victory.

Type: Problem-Solving Task

This is the first and most basic problem in a series of seven problems, all set in the context of a classroom election. Students are given a ratio and total number of voters and are asked to determine the number of votes received by each candidate.

Type: Problem-Solving Task

This is the second in a series of tasks that are set in the context of a classroom election. It requires students to understand what ratios are and apply them in a context. The simple version of this question just asked how many votes each gets. This has the extra step of asking for the difference between the votes.

Type: Problem-Solving Task

This is the fourth in a series of tasks about ratios set in the context of a classroom election. Given only a ratio, students are asked to determine the fractional difference between votes received and votes required.

Type: Problem-Solving Task

Students are given a context and a dotplot and are asked a number of questions regarding shape, center, and spread of the data.

Type: Problem-Solving Task

Students are given a context and a series of questions and are asked to identify whether each question is statistical and to provide their reasoning. Students are asked to compose an original statistical question for the given context.

Type: Problem-Solving Task

Using the information provided, create an appropriate graphical display and answer the questions regarding shape, center and variability.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

The purpose of this task is for students to translate between measurements given in a scale drawing and the corresponding measurements of the object represented by the scale drawing. If used in an instructional setting, it would be good for students to have an opportunity to see other solution methods, perhaps by having students with different approaches explain their strategies to the class. Students who can only solve this by first converting the linear measurements will have a hard time solving problems where only area measures are given.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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?

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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?

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

In this activity, the student is asked to solve a variety of equations (one solution, infinite solutions, no solution) in the traditional algebraic manner and to use pictures of a pan balance to show the solution process.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

It is possible to say a lot about the solution to an equation without actually solving it, just by looking at the structure and operations that make up the equation. This exercise turns the focus away from the familiar "finding the solution" problem to thinking about what it really means for a number to be a solution of an equation.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

The purpose of this task is for students to interpret two distance-time graphs in terms of the context of a bicycle race. There are two major mathematical aspects to this: interpreting what a particular point on the graph means in terms of the context and understanding that the "steepness" of the graph tells us something about how fast the bicyclists are moving.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

This task lets students explore the differences between linear and non-linear functions. By contrasting the two, it reinforces properties of linear functions.

Type: Problem-Solving Task

The primary purpose of this task is to elicit common misconceptions that arise when students try to model situations with linear functions. This task, being multiple choice, could also serve as a quick assessment to gauge a class' understanding of modeling with linear functions.

Type: Problem-Solving Task

This is a simple task about interpreting the graph of a function in terms of the relationship between quantities that it represents.

Type: Problem-Solving Task

In this task students draw the graphs of two functions from verbal descriptions. Both functions describe the same situation but changing the viewpoint of the observer changes where the function has output value zero. This small twist forces the students to think carefully about the interpretation of the dependent variable. This task could be used in different ways: To generate a class discussion about graphing. As a quick assessment about graphing, for example during a class warm-up. To engage students in small group discussion.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Students will just be learning about similarity in this grade, so they may not recognize that it is needed in this context. Teachers should be prepared to give support to students who are struggling with this part of the task. To simplify the task, the teacher can just tell the students that based on the slant of the truncated conical cup, the complete cone would be 14 in tall and the part that was sliced off was 10 inches tall. (See solution for an explanation.) There is a worthwhile discussion to be had about parts (c) and (e). The percentage increase is smaller for the snow cones than it was for the juice treats. The snow cones have volume which is equal to those of the juice treats plus the volume of the dome, which is the same in both cases. Adding the same number to two numbers in a ratio will always make their ratio closer to one, which in this case means that the ratio - and thus percentage increase - would be smaller.

Type: Problem-Solving Task

Students' first experience with transformations is likely to be with specific shapes like triangles, quadrilaterals, circles, and figures with symmetry. Exhibiting a sequence of transformations that shows that two generic line segments of the same length are congruent is a good way for students to begin thinking about transformations in greater generality.

Type: Problem-Solving Task

This task has two goals: first to develop student understanding of rigid motions in the context of demonstrating congruence. Secondly, student knowledge of reflections is refined by considering the notion of orientation in part (b). Each time the plane is reflected about a line, this reverses the notions of ''clockwise'' and ''counterclockwise.''

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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?

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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

Type: Problem-Solving Task

In a poll of Mr. Briggs's math class, 67% of the students say that math is their favorite academic subject. The editor of the school paper is in the class, and he wants to write an article for the paper saying that math is the most popular subject at the school. Explain why this is not a valid conclusion and suggest a way to gather better data to determine what subject is most popular.

Type: Problem-Solving Task

In this task, students are able to conjecture about the differences and similarities in the two groups from a strictly visual perspective and then support their comparisons with appropriate measures of center and variability. This will reinforce that much can be gleaned simply from visual comparison of appropriate graphs, particularly those of similar scale.

Type: Problem-Solving Task

The purpose of this task is to provide students with the opportunity to determine experimental probabilities by collecting data. The cylindrical objects used in this task typically have three different resting positions but not all of these may be equally likely and some may be extremely unlikely or impossible when the object is tossed. Furthermore, obtaining the probabilities of the outcomes is perhaps only possible through the use of long-run relative frequencies. This is because these cylinders do not have the same types of symmetries as objects that are often used as dice, such as cubes or tetrahedrons, where each outcome is equally likely.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

This resource involves a simple data-gathering activity which furnishes data that students organize into a table. They are then asked to refer to the data and determine the probability of various outcomes.

Type: Problem-Solving Task

This task introduces the fundamental statistical ideas of using data summaries (statistics) from random samples to draw inferences (reasoned conclusions) about population characteristics (parameters). In the task built around an election poll scenario, the population is the entire seventh grade class, the unknown characteristic (parameter) of interest is the proportion of the class members voting for a specific candidate, and the sample summary (statistic) is the observed proportion of voters favoring the candidate in a random sample of class members. Variation 2 leads students through a physical simulation for generating sample proportions by sampling, and re-sampling, marbles from a box.

Type: Problem-Solving Task

This task introduces the fundamental statistical ideas of using data summaries (statistics) from random samples to draw inferences (reasoned conclusions) about population characteristics (parameters). There are two important goals in this task: seeing the need for random sampling and using randomization to investigate the behavior of a sample statistic. These introduce the basic ideas of statistical inference and can be accomplished with minimal knowledge of probability.

Type: Problem-Solving Task

As the standards in statistics and probability unfold, students will not yet know the rules of probability for compound events. Thus, simulation is used to find an approximate answer to these questions. In fact, part b would be a challenge to students who do know the rules of probability, further illustrating the power of simulation to provide relatively easy approximate answers to wide-ranging problems.

Type: Problem-Solving Task

This task is intended as a classroom activity. Students pool the results of many repetitions of the random phenomenon (rolling dice) and compare their results to the theoretical expectation they develop by considering all possible outcomes of rolling two dice. This gives them a concrete example of what we mean by long term relative frequency.

Type: Problem-Solving Task

The purpose of this task is for students to compute the theoretical probability of a compound event. Teachers may wish to emphasize the distinction between theoretical and experimental probabilities for this problem. For students learning to distinguish between theoretical and experimental probability, it would be good to find an experimental probability either before or after students have calculated the theoretical probability.

Type: Problem-Solving Task

The purpose of this task is for students to compute the theoretical probability of a seating configuration. There are 24 possible configurations of the four friends at the table in this problem. Students could draw all 24 configurations to solve the problem but this is time consuming and so they should be encouraged to look for a more systematic method.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

Type: Problem-Solving Task

This task provides us with the opportunity to see how the mathematical ideas embedded in the standards and clusters mature over time. The task "Uses facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure (MAFS.7.G.2.5)" except that it requires students to know, in addition, something about parallel lines, which students will not see until 8th grade. As a result, this task is especially good at illustrating the links between related standards across grade levels.

Type: Problem-Solving Task

The purpose of this task is to give students practice working the formulas for the volume of cylinders, cones and spheres, in an engaging context that provides and opportunity to attach meaning to the answers.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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."

Type: Problem-Solving Task

When students plot irrational numbers on the number line, it helps reinforce the idea that they fit into a number system that includes the more familiar integer and rational numbers. This is a good time for teachers to start using the term "real number line" to emphasize the fact that the number system represented by the number line is the real numbers. When students begin to study complex numbers in high school, they will encounter numbers that are not on the real number line (and are, in fact, on a "number plane"). This task could be used for assessment, or if elaborated a bit, could be used in an instructional setting.

Type: Problem-Solving Task

Students should think of different ways the cylindrical containers can be set up in a rectangular box. Through the process, students should realize that although some setups may seem different, they result in a box with the same volume. In addition, students should come to the realization (through discussion and/or questioning) that the thickness of a cardboard box is very thin and will have a negligible effect on the calculations.

Type: Problem-Solving Task

This task is ideally suited for instruction purposes where students can take their time and develop several of the Mathematical Practice standards, as the mathematical content is directly related to, but somewhat exceeds, the content of standard MAFS.8.G.1.5 on sums of angles in triangles. Careful analysis of the angles requires students to construct valid arguments (MAFS.K12.MP.3.1) using abstract and quantitative reasoning (MAFS.K12.MP.2.1). Producing the picture in part (c) helps students identify a common mathematical argument repeated multiple times (MAFS.K12.MP.8.1). If students use pattern blocks in order to develop the intuition for decomposing the hexagon into triangles, then this is also an example of MAFS.K12.MP.5.1.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Students are given a pair of numbers. They are asked to determine which is larger, and then justify their answer. The numbers involved are rational numbers and common irrational numbers, such π and square roots. This task can be used to either build or assess initial understandings related to rational approximations of irrational numbers.

Type: Problem-Solving Task

This task asks the student to gather data on whether classmates play an instrument and/or participate in a sport, summarize the data in a table and decide whether there is an association between playing a sport and playing an instrument. Finally, the student is asked to create a graph to display any association between the variables.

Type: Problem-Solving Task

Students are asked to examine data given in table format and then calculate either row percentages or column percentages and state a conclusion about the meaning of the data. Either calculation is appropriate for the solution since there is no clear relationship between the variables. Whether the student sees a strong association or not is less important than whether his or her answer uses the data appropriately and demonstrates understanding that an association means the distribution of favorite subject is different for 7th graders and 8th graders.

Type: Problem-Solving Task

Students are asked to examine a scatter plot and then interpret its meaning. Students should identify the form of the relationship (linear, curved, etc.), the direction or correlation (positive or negative), any specific outliers, the strength of the relationship between the two variables, and any other relevant observations.

Type: Problem-Solving Task

In this resource, real-world bivariate data is displayed in a scatter plot. The equation of the linear function which models the relationship between the two variables is provided, and it is graphed on the scatter plot. Students are asked to use the model to interpret the data and to make predictions.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

In this task students interpret two graphs that look the same but show very different quantities. The first graph gives information about how fast a car is moving while the second graph gives information about the position of the car. This problem works well to generate a class or small group discussion. Students learn that graphs tell stories and have to be interpreted by carefully thinking about the quantities shown.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Students are asked a series of questions involving a fraction and a whole number within the context of a recipe. Students are asked to solve a problem using both a visual model and the standard algorithm.

Type: Problem-Solving Task

Students are asked to solve a distance problem involving fractions. The purpose of this task is to help students extend their understanding of division of whole numbers to division of fractions, and given the simple numbers used, it is most appropriate for students just learning about fraction division because it lends itself easily to a pictorial solution.

Type: Problem-Solving Task

The purpose of this task is for students to solve problems involving multiplication and division of decimals in the real-world context of setting financial goals. The focus of the task is on modeling and understanding the concept of setting financial goals, so fluency with the computations will allow students to focus on other aspects of the task.

Type: Problem-Solving Task

In this problem-solving task students are challenged to apply their understanding of linear relationships to determine the amount of chicken and steak needed for a barbecue, which will include creating an equation, sketching a graph, and interpreting both. This resource also includes standards alignment commentary and annotated solutions.

Type: Problem-Solving Task

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).

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Students are asked to solve a real-world problem involving common multiples.

Type: Problem-Solving Task

Students are asked to use fractions to determine how many hours it will take a car to travel a given distance.

Type: Problem-Solving Task

Students are asked to use fractions to determine how long a video game can be played.

Type: Problem-Solving Task

The purpose of this task is to have students convert multiple currencies to answer the problem. Students may find the CDN abbreviation for Canada confusing. Teachers may need to explain the fact that money in Canada is also called dollars, so to distinguish them, we call them Canadian dollars.

Type: Problem-Solving Task

Use the information provided to find out what percentage of Dana's lot won't be covered by the house.

Type: Problem-Solving Task

This task asks the students to solve a real-world problem involving unit rates (data per unit time) using units that many teens and pre-teens have heard of but may not know the definition for. While the computations involved are not particularly complex, the units will be abstract for many students. The first solution relies more on reasoning about the meaning of multiplication and division, while the second solution uses units to help keep track of the steps in the solution process.

Type: Problem-Solving Task

In this problem-solving task students are challenged to apply their understanding of linear relationships to determine the amount of chicken and steak needed for a barbecue, which will include creating an equation, sketching a graph, and interpreting both. This resource also includes standards alignment commentary and annotated solutions.

Type: Problem-Solving Task

This problem-solving task asks students to find a linear function that models something in the real world. After finding the equation of the linear relationship between the depth of the water and the distance across the channel, students have to verbalize the meaning of the slope and intercept of the line in the context of this situation. Commentary on standards alignment and illustrated solutions are also included.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

The student is asked to perform operations with numbers expressed in scientific notation to decide whether 7% of Americans really do eat at Giantburger every day.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Students are asked to use knowledge of rates and ratios to answer a series of questions involving time, distance, and speed.

Type: Problem-Solving Task

Students are asked to write complete sentences to describe ratios for the context.

Type: Problem-Solving Task

The purpose of the task is for students to compare signed numbers in a real-world context.

Type: Problem-Solving Task

The purpose of this task is to help students explore the meaning of fraction division and to connect it to what they know about whole-number division. Students are asked to explain why the quotient of two fractions with common denominators is equal to the quotient of the numerators of those fractions.

Type: Problem-Solving Task

This task builds on a fifth grade fraction multiplication task, "Drinking Juice." This task uses the identical context, but asks the corresponding "Number of Groups Unknown" division problem. See "Drinking Juice, Variation 3" for the "Group Size Unknown" version.

Type: Problem-Solving Task

Students are asked to solve a fraction division problem using a visual model and the standard algorithm.

Type: Problem-Solving Task

Students are asked to solve problems from context by using multiplication or division of decimals.

Type: Problem-Solving Task

This instructional task requires that the students model each problem with some type of fractions manipulatives or drawings. This could be pattern blocks, student or teacher-made fraction strips, or commercially produced fraction pieces. At a minimum, students should draw pictures of each. The above problems are meant to be a progression which require more sophisticated understandings of the meaning of fractions as students progress through them.

Type: Problem-Solving Task

The purpose of this task is for students to get a better understanding of the relative positions and values of positive and negative numbers.

Type: Problem-Solving Task

The purpose of this task is for students to apply their knowledge of integers in a real-world context.

Type: Problem-Solving Task

Students are asked to add or subtract decimals to solve problems in context.

Type: Problem-Solving Task

Students are asked to write and solve an equation in one variable to answer a real world question.

Type: Problem-Solving Task

In this task students use different representations to analyze the relationship between two quantities and to solve a real world problem. The situation presented provides a good opportunity to make connections between the information provided by tables, graphs and equations. In the later part of the problem, the numbers are big enough so that using the formula is the most efficient way to solve the problem; however, creative use of the table or graph will also work.

Type: Problem-Solving Task

This task asks students to find equivalent expressions by visualizing a familiar activity involving distance. The given solution shows some possible equivalent expressions, but there are many variations possible.

Type: Problem-Solving Task

Students are asked to use properties of operations to match expressions that are equivalent and to write equivalent expressions for any expressions that do not have a match.

Type: Problem-Solving Task

In this task students are asked to write an equation to solve a real-world problem.

Type: Problem-Solving Task

Students are asked to write and graph two inequalities described in context: one discrete and one continuous.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

Students are asked to write an equation with one variable in order to find the distance walked.

Type: Problem-Solving Task

This purpose of this task is to help students understand the absolute value of a number as its distance from 0 on the number line. The context is not realistic, nor is meant to be; it is a thought experiment to help students focus on the relative position of numbers on the number line.

Type: Problem-Solving Task

Students are asked to determine if two different ratios are both appropriate for the same context.

Type: Problem-Solving Task

Given a ratio, students are asked to determine how much of each ingredient is needed to make concrete.

Type: Problem-Solving Task

This problem provides an interesting geometric context to work on the notion of percent. Two different methods for analyzing the geometry are provided: the first places the two squares next to one another and then moves one so that they overlap. The second solution sets up an equation to find the overlap in terms of given information which reflects the mathematical ideas described in cluster MAFS.6.EE.2 - Reason about and solve one-variable equations and inequalities.

Type: Problem-Solving Task

Students are asked to use a given ratio to determine if two different interpretations of the ratio are correct and to determine the maximum quantity that could be purchased within a given context.

Type: Problem-Solving Task

Students are asked apply knowledge of ratios to answer several questions regarding speed, distance and time.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

This site uses visual models to better understand what is actually happening when students multiply and divide fractions. Using area models -- one that superimposes squares that are partitioned into the appropriate number of regions, and shaded as needed -- students multiply, divide, and translate the processes to decimals. The lesson uses an interactive simulation that allows students to create their own area models and is embedded with problems throughout for students to solve.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

## Student Center Activities

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

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

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

Type: Tutorial

In this tutorial students are asked to find missing angle measures from a variety of examples.

Type: Tutorial

In this example, students will use algebra to find the measure of two angles whose sum equals 90 degrees, better known as complementary angles.

Type: Tutorial

In this tutorial, students are asked to prove two angles congruent when given limited information. Students need to have a foundation of parallel lines, transversals and triangles before viewing this video.

Type: Tutorial

This video demonstrates finding the volume and surface area of a cylinder.

Type: Tutorial

This video introduces the concept of rigid transformation and congruent figures.

Type: Tutorial

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

Type: Tutorial

This video shows testing for similarity through transformations.

Type: Tutorial

This video explains the formula for volume of a cone and applies the formula to solve a problem.

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

Let's use algebra to find the measure of two complementary angles.

Type: Tutorial

Let's use algebra to find the measure of supplementary angles, whose sum is 180 degrees.

Type: Tutorial

This tutorial shows students how to find the distance between lines using the Pythagorean Theorem. This video makes a connection between the distance formula and the Pythagorean Theorem.

Type: Tutorial

This video gives the proof of sum of measures of angles in a triangle. This video is beneficial for both Algebra and Geometry students.

Type: Tutorial

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

Type: Tutorial

This video demonstrates a system of equations with no solution.

Type: Tutorial

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

Type: Tutorial

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

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

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

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

In this video, you will determine if the situation is linear or non-linear by finding the rate of change between cooordinates. You will check your work by graphing the coordinates given.

Type: Tutorial

In this tutorial, students will compare linear functions from a graph. Students should have an understanding of slope and rate of change before reviewing this tutorial.

Type: Tutorial

This tutorial shows how to compare linear functions that are presented in both a table and graph. Students should have an understanding of rate of change before viewing this video.

Type: Tutorial

Students will compare linear functions presented in a graph and in a table. Students should have a strong understanding of rate of change before viewing this tutorial.

Type: Tutorial

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

In this video, you will practice finding the slope of a line from data in a table, and interpret what the slope means in the problem.

Type: Tutorial

In this video, you will use a linear graph to determine the y-intercept (starting point) and slope (rate of change), as well as interpret what these mean in the given scenario.

Type: Tutorial

In this tutorial, you will look at several real-world examples of linear graphs and interpret the relationship between the two variables.

Type: Tutorial

In this video, you will practice writing the slope-intercept form for a line, given a table of x and y values.

Type: Tutorial

Students will learn how to find and graph the x and y intercepts from an equation written in standard form.

Type: Tutorial

Students will learn how to find the x and y intercepts from an equation in standard form.

Type: Tutorial

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

Type: Tutorial

This tutorial shows students how to find the y inercept from a table.

Type: Tutorial

Given two points on a line, you will find the slope and the y-intercept. You will then write the equation of the line in slope-intercept form.

Type: Tutorial

Students will learn how to graph a linear equation using a table. Students will not be required to graph from slope-intercept form, although they will convert the equation from standard form to slope-intercpet form before they create the table.

Type: Tutorial

Given the slope of a line and a point on the line, you will write the equation of the line in slope-intercept form.

Type: Tutorial

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

Students will learn how to determine an equation by checking solutions. Students will be given a table and 4 linear equations and they will have to determine which equation created the table.

Type: Tutorial

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

Type: Tutorial

This tutorial shows how to find the slope from two ordered pairs. Students will see what happens to the slope of a horizontal line.

Type: Tutorial

In this video, you will practice writing the equations of lines in slope-intercept form from graphs. You will then practice graphing lines from equations in slope-intercept form.

Type: Tutorial

In this tutorial, you will use your knowledge about similar triangles, as well as parallel lines and transversals, to prove that the slope of any given line is constant.

Type: Tutorial

This tutprial shows how to graph a line in slope-intercept form.

Type: Tutorial

This tutorial shows an example of finding the slope between two ordered pairs. Slope is presented as rise/run, the change in y divided by the change in x and also as m.

Type: Tutorial

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

In this video, you will practice comparing an irrational number to a percent. First you will try it without a calculator. Then you will check your answer using a calculator.

Type: Tutorial

In this video, you will learn how to approximate a square root to the hundredths place.

Type: Tutorial

In this video, you will practice approximating square roots of numbers that are not perfect squares. You will find the perfect square below and above to approximate the value of the square root between two whole numbers.

Type: Tutorial

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

Type: Tutorial

In this video, you will practice describing the shape of distributions as skewed left, skewed right, or symmetrical.

Type: Tutorial

In this video, you will see two ways to find the Mean Absolute Deviation of a data set.

Type: Tutorial

Use the Distributive Property while solving equations with variables on both sides.

Type: Tutorial

Students will learn how to solve an equation with variables on both sides. This tutorial shows a final answer expressed as an improper fraction and mixed number.

Type: Tutorial

This video shows how to solve the equation (3/4)x + 2 = (3/8)x - 4 using the Distributive Property.

Type: Tutorial

This video shows how to solve an equation involving the Distributive Property.

Type: Tutorial

This example involves a variable in the denominator on both sides of the equation.

Type: Tutorial

This video discusses exponent properties involving products.

Type: Tutorial

Students will learn how to solve an equation with variables on both sides. Students will also learn how to distribute and combine like terms.

Type: Tutorial

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

Type: Tutorial

Students will learn the difference between rational and irrational numbers.

Type: Tutorial

This video demonstrates multiplying in scientific notation.

Type: Tutorial

This example demonstrates mathematical operations with scientific notation used to solve a word problem.

Type: Tutorial

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

Type: Tutorial

This video demonstrates a scientific notation word problem involving division.

Type: Tutorial

Students will learn that non-zero numbers to the zero power equals one. They will also learn that zero to any positive exponent equals zero. They will then investigate what happens when you have zero to the zero power.

Type: Tutorial

This is an example showing how to simplify an expression into scientific notation.

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

This video demonstrates several examples of finding probability of random events.

Type: Tutorial

This video discusses the limits of probability as between 0 and 1.

Type: Tutorial

This video compares theoretical and experimantal probabilities and sources of possible discrepancy.

Type: Tutorial

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

Type: Tutorial

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

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

Type: Tutorial

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

Type: Tutorial

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

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

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

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

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

Predict the number of times a spinner will land on a given outcome.

Type: Tutorial

This video demonstrates development and use of a probability model.

Type: Tutorial

This video explores how to create sample spaces as tree diagrams, lists and tables.

Type: Tutorial

This video shows how to use a sample space diagram to find probability.

Type: Tutorial

This video shows an example of using a tree diagram to find the probability of a compound event.

Type: Tutorial

This video uses knowledge of vertical angles to solve for the variable and the angle measures.

Type: Tutorial

This video uses facts about supplementary and adjacent angles to introduce vertical angles.

Type: Tutorial

This video demonstrates solving a word problem involving angle measures.

Type: Tutorial

This video discusses constructing a right isosceles triangle with given constraints and deciding if the triangle is unique.

Type: Tutorial

This video demonstrates drawing a triangle when the side lengths are given.

Type: Tutorial

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

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

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

In this tutorial, you will compare rational numbers using a number line.

Type: Tutorial

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

Type: Tutorial

In this video, you will practice using arithmetic properties with integers to determine if expressions are equivalent.

Type: Tutorial

You will discover rules to help you determine the sign of an exponential expression with a base of -1.

Type: Tutorial

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

The focus of this video is to help you understand the core concepts of arithmetic mean, median, and mode.

Type: Tutorial

This video shows how to find the value of a missing piece of data if you know the mean of the data set.

Type: Tutorial

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

Type: Tutorial

This video teaches about combining like terms in linear equations.

Type: Tutorial

This video demonstrates how to construct a box and whisker plot.

Type: Tutorial

Students will interpret data presented in a box and whisker plot.

Type: Tutorial

In this tutorial, you will apply what you know about multiplying negative numbers to determine how negative bases with exponents are affected and what patterns develop.

Type: Tutorial

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

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

Type: Tutorial

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

Type: Tutorial

We will understand the difference between supplementary angles and complementary angles, by using the given measurements of angles.

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

In this tutorial, you will see how mixed numbers can be divided.

Type: Tutorial

This tutorial demonstrates how the area of an irregular geometric shape may be determined by decomposition to smaller familiar shapes.

Type: Tutorial

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

Another way of finding the volume of a rectangular prism involves dividing it into fractional cubes, finding the volume of one, and then multiplying that area by the number of cubes that fit into the rectangular prism. Watch this explanation.

Type: Tutorial

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

Type: Tutorial

This video shows how to solve a word problem involving rectangular prisms.

Type: Tutorial

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

Type: Tutorial

This video demonstrates how to construct nets for 3-D shapes.

Type: Tutorial

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

Students will graph the given coordinates of three of the polygon vertices, then locate and graph the fourth vertex.

Type: Tutorial

This video demonstrates using a net to find surface area.

Type: Tutorial

In this example students are given the coordinates of the vertices and asked to construct the resulting polygon, specifically a quadrilateral.

Type: Tutorial

In this video, we organize data into frequency tables and dot plots (sometimes called line plots).

Type: Tutorial

Solve a multi-step word problem in the context of a cab fare.

Type: Tutorial

In this example, you determine the volume of frozen water and express the answer as a fraction.

Type: Tutorial

Learn how to create histograms, which summarize data by sorting it into groups (buckets).

Type: Tutorial

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

Type: Tutorial

Here's an introduction to basic algebraic equations of the form ax = b. Remember that you can check to see if you have the right answer by substituting it for the variable!

Type: Tutorial

Learn how to solve equations in one step by multiplying or dividing a number on both sides. These problems involve decimals and fractions.

Type: Tutorial

Students will solve the inequality and graph the solution.

Type: Tutorial

Learn how to solve a word problem by writing an equation to model the situation. In this video, we use the linear equation 210(t-5) = 41,790.

Type: Tutorial

This tutorial shows a word problem in which students will find the dimensions of a garden given only the perimeter. Students will create an equation to solve.

Type: Tutorial

This example demonstrates how to solve an equation expressed in the form ax + b = c.

Type: Tutorial

This video shows how to solve an equation by isolating the variable in the numerator.

Type: Tutorial

A solution to an inequality makes that inequality true. Learn how to test if a certain value of a variable makes an inequality true.

Type: Tutorial

Students will practice two step equations, some of which require combining like terms and using the distributive property.

Type: Tutorial

A solution to an equation makes that equation true. Learn how to test if a certain value of a variable makes an equation true.

Type: Tutorial

This video demonstrates how to write and solve a one-step addition equation.

Type: Tutorial

To find the value of a variable, you have to get it on one side of the equation alone. To do that, you'll need to do something to BOTH sides of the equation.

Type: Tutorial

This video provides a conceptual explanation of why one needs to divide both sides of an equation to solve for a variable.

Type: Tutorial

In an equation with 2 variables, we will be able to determine which is the dependent variable, and which is the independent variable.

Type: Tutorial

Learn how to write basic algebraic expressions.

Type: Tutorial

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

Type: Tutorial

Learn how to write inequalities to model real-world situations.

Type: Tutorial

Given a graph, we will be able to find the equation it represents.

Type: Tutorial

Learn how to write simple algebraic expressions.

Type: Tutorial

This video shows how to solve a two step equation. It begins with the concept of equality, what is done to one side of an equation, must be done to the other side of an equation.

Type: Tutorial

Learn how to write basic expressions with variables to portray situations described in word problems.

Type: Tutorial

Learn how to apply the distributive law of multiplication over addition and why it works. This is sometimes just called the distributive law or the distributive property.

Type: Tutorial

Learn how to apply the distributive property of multiplication over subtraction. This is sometimes just called the distributive property or distributive law.

Type: Tutorial

Learn how to apply the distributive property to algebraic expressions.

Type: Tutorial

This video demonstrates solving word problems involving the coordinate plane.

Type: Tutorial

The focus here is understanding that a variable is just a symbol that can represent different values in an expression.

Type: Tutorial

Learn how to evaluate an expression with variables using a technique called substitution.

Type: Tutorial

This video demonstrates evaluating expressions with two variables.

Type: Tutorial

Explore how the value of an algebraic expression changes as the value of its variable changes.

Type: Tutorial

In this example, we have a formula for converting a Celsius temperature to Fahrenheit.

Type: Tutorial

Students will simplify an expression by combining like terms.

Type: Tutorial

Students will plot an ordered pair on the x (horizontal) axis and y (vertical) axis of the coordinate plane.

Type: Tutorial

This tutorial is a good explanation on how to combine like terms in algebra.

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

This video demonstrates the prime factorization method to find the lcm (least common multiple).

Type: Tutorial

Students will become familiar with the x/y coordinate plane, both from the perspective of plotting points and interpreting the placement of points on a plane.

Type: Tutorial

This video contains several examples of plotting coordinate pairs and identifying their quadrant.

Type: Tutorial

This video discusses the negative sign as meaning "opposite."

Type: Tutorial

We're mixing it up by placing both fractions and decimals on the same number line. Great practice because you need to move effortlessly between the two.

Type: Tutorial

Is -40 bigger than -10? When ordering negative numbers from least to greatest, be careful that you don't get hung up on the "amount" of the number. Think about what that negative sign really means!

Type: Tutorial

In this tutorial, you will learn how to order rational numbers using a number line.

Type: Tutorial

In this tutorial you will compare the absolute value of numbers using the concepts of greater than (>), less than (<), and equal to (=).

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

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

This video guides you through comparisons of values, including opposites.

Type: Tutorial

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

Type: Tutorial

This video demonstrates sorting values including absolute value from least to greatest using a number line.

Type: Tutorial

This video demonstrates evaluating inequality statements, some involving absolute value, using a number line.

Type: Tutorial

In simple addition we learned to add all the numbers together to get a sum. In algebra, numbers are sometimes attached to variables and we need to make sure that the variables are alike before we add the numbers. This tutorial is an introduction to combining like terms.

Type: Tutorial

This video demonstrates solving absolute value inequality statements.

Type: Tutorial

Students will evaluate expressions using the order of operations.

Type: Tutorial

Practice interpreting absolute value in a real-life situation.

Type: Tutorial

Students will learn how to identify the four quadrants in the coordinate plane.

Type: Tutorial

This video uses a number line to describe the opposite of a number.

Type: Tutorial

Work through another challenging order of operations example with only positive numbers.

Type: Tutorial

Work through a challenging order of operations example with only positive numbers.

Type: Tutorial

This video will show how to evaluate expressions with exponents using the order of operations.

Type: Tutorial

This video demonstrates dividing two decimal numbers.

Type: Tutorial

This video demonstrates dividing fractions as multiplying by the reciprocal.

Type: Tutorial

This video demonstrates dividing a whole number by a fraction by multiplying by the reciprocal.

Type: Tutorial

This video portrays a proof of the formula for area of a parallelogram.

Type: Tutorial

This video demonstrates how to evaluate expressions with whole number exponents.

Type: Tutorial

A trapezoid is a type of quadrilateral with one set of parallel sides. Here we explain how to find its area.

Type: Tutorial

This video demonstrates how to multiply two decimal numbers.

Type: Tutorial

Students will be able to find the area of a triangle in a coordinate grid. The formula for the area of a triangle is given in this tutorial.

Type: Tutorial

Students will learn the basics of finding the perimeter and area of squares and rectangles.

Type: Tutorial

This video demonstrates adding decimal numbers to solve a word problem.

Type: Tutorial

This video shows an example of subtracting with digits up to the thousandths place.

Type: Tutorial

Just like when add, be sure you align decimals before subtracting.

Type: Tutorial

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

Type: Tutorial

Practice substituting positive and negative values for variables.

Type: Tutorial

Learn how to add 9.087 to 15.31. Be careful to use place value!

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

Let's solve this word problem using what we know about equivalent ratios.

Type: Tutorial

In this example, we are given a ratio and then asked to apply that ratio to solve a problem. No problem!

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

You are asked to find the percent when given the part and the whole.

Type: Tutorial

This video demonstrates how to find percent of a whole number.

Type: Tutorial

You're asked to find the whole when given the part and the percent.

Type: Tutorial

It's nice to practice conversion problems, but how about applying our new knowledge of percentages to a real life problem like recycling? Hint: don't forget your long division!

Type: Tutorial

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

Type: Tutorial

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

Evaluating Expressions with Two Variables

Type: Tutorial

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

This video demonstrates solving a unit price problem using equivalent ratios.

Type: Tutorial

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

Type: Tutorial

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

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

Type: Tutorial

This video talks about what percent really means by looking at a 10 by 10 grid.

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

This resource will allow students to have a good understanding about vertical, adjacent and linear pairs of angles.

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

The distributive property states that the terms of addition or subtraction statements within parentheses may be separately multiplied by a value outside the parentheses. In this tutorial, students will learn the distributive property, which is very helpful with mental math calculations and solving equations.

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

This video tutorial shows examples of writing expressions in simplified form and evaluating expressions.

Type: Tutorial

This tutorial reviews the mathematical order of operations and reminds students why common memory tricks might be misleading.

Type: Tutorial

Equations of the form *y* = *mx* describe lines in the Cartesian plane which pass through the origin. The fact that many functions are linear when viewed on a small scale, is important in branches of mathematics such as calculus.

Type: Tutorial

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

This video provides assistance with understanding direct and inverse variation.

Type: Tutorial

This short video explains how to solve multi-step equations with variables on both sides and why it is necessary to complete the same steps on both sides of the equation.

Type: Tutorial

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

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

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

The Cartesian Coordinate system, formed from the Cartesian product of the real number line with itself, allows algebraic equations to be visualized as geometric shapes in two or three dimensions.

Type: Tutorial

Scatterplots are used to visualize the relationship between two quantitative variables in a binary relation. As an example, trends in the relationship between the height and weight of a group of people could be graphed and analyzed using a scatter plot.

Type: Tutorial

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

Linear equations of the form y=mx+b can describe any non-vertical line in the cartesian plane. The constant m determines the line's slope, and the constant b determines the y intercept and thus the line's vertical position.

Type: Tutorial

Scientific notation is used to conveniently write numbers that require many digits in their representations. How to convert between standard and scientific notation is explained in this tutorial.

Type: Tutorial

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

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

Take a look at the logic behind the associative and distributive properties of multiplication.

Type: Tutorial

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

Type: Tutorial

Number systems evolved from the natural "counting" numbers, to whole numbers (with the addition of zero), to integers (with the addition of negative numbers), and beyond. These number systems are easily understood using the number line.

Type: Tutorial

The commutative property is common to the operations of both addition and multiplication and is an important property of many mathematical systems.

Type: Tutorial

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

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

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

This resource helps the user learn the three primary colors that are fundamental to human vision, learn the different colors in the visible spectrum, observe the resulting colors when two colors are added, and learn what white light is. A combination of text and a virtual manipulative allows the user to explore these concepts in multiple ways.

Type: Tutorial

The user will learn the three primary subtractive colors in the visible spectrum, explore the resulting colors when two subtractive colors interact with each other and explore the formation of black color.

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

The video demonstrates rewriting given numbers in a common format (as decimals), so they can be compared and ordered.

Type: Tutorial

Introduction to solving one variable multiplication equations of the form px = q.

Type: Tutorial

## Video/Audio/Animations

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

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

Type: Video/Audio/Animation

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

When working with fractions, divisions can be converted to multiplication by the divisor's reciprocal. This chapter explains why.

Type: Video/Audio/Animation

This 5-minute video provides an example of how to solve a problem using a trend line to estimate data through a problem called, "Smoking in 1945."

Type: Video/Audio/Animation

This 6-minute video provides an example of how to work with compound probability of independent events through the example of flipping a coin. If you flip a coin and it lands on heads, is the next flip more likely to be tails? Or are those events independent?

Type: Video/Audio/Animation

This 8-minute video provides an introduction to the concept of probability through the example of flipping a coin and rolling a die.

Type: Video/Audio/Animation

This short video provides a clear explanation why we perform the same steps on each side of an equation when solving for the variable/unknown.

Type: Video/Audio/Animation

This short video provides a clear explanation about the "why" of performing the same steps on each side of an equation when solving for the variable/unknown.

Type: Video/Audio/Animation

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

Type: Video/Audio/Animation

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

Exponentiation can be understood in terms of repeated multiplication much like multiplication can be understood in terms of repeated addition. Properties of multiplication and division of exponential expressions with the same base are derived.

Type: Video/Audio/Animation

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

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

Type: Video/Audio/Animation

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

Type: Video/Audio/Animation

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

Type: Video/Audio/Animation

Percentages are one method of describing a fraction of a quantity. the percent is the numerator of a fraction whose denominator is understood to be one-hundred.

Type: Video/Audio/Animation

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

Type: Video/Audio/Animation

Ratio errors confuse one of the coaches as two teams face off in an epic dodgeball tournament. See how mathematical techniques such as tables, graphs, measurements and equations help to find the missing part of a proportion.

Atlantean Dodgeball addresses number and operations standards, the algebra standard, and the process standard, as established by the National Council of Teachers of Mathematics (NCTM). It guides students in:

- Understanding and using ratios and proportions to represent quantitative relationships.
- Relating and comparing different forms of representation for a relationship.
- Developing, analyzing, and explaining methods for solving problems involving proportions, such as scaling and finding equivalent ratios.
- Representing, analyzing, and generalizing a variety of patterns with tables, graphs, words, and, when possible, symbolic rules.

Type: Video/Audio/Animation

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

Type: Video/Audio/Animation

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

Type: Video/Audio/Animation

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

Type: Video/Audio/Animation

## Virtual Manipulatives

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

This virtual manipulative is a probability simulation tool which will help the learners to understand the concepts of experimental and theoretical probability.

Type: Virtual Manipulative

The purpose of this manipulative is to help students recognize that (1) unusual events do happen, and (2) it may take a longer time for some of them to happen. The letters are drawn at random from the beginning of Hamlet's soliloquy, "To be, or not to be." Any word made from those letters (such as TO) can be entered in the box. When the *start* is pressed, letters are drawn and recorded. The process continues until the word appears.

Type: Virtual Manipulative

This drag and drop Venn diagram simulation gives students the opportunity to solve a mathematical problem based on number properties using a range of different Venn diagrams. There are five different levels involving a range of multiples and simply odds and evens. The three core layouts cover simple separate sets, two intersecting sets, and a three way intersecting Venn Diagram. The odds and evens layout is limited to two intersecting sets, of course.

Type: Virtual Manipulative

This virtual manipulative can be used demonstrate random probability and to teach about chance and random choices. Use this free, fully customizable, online spinner to create probability scenarios involving numerous choices, or create advanced, unevenly split spinners to demonstrate and model real life scenarios.

This spinner also incorporates a bar graph to record and model the outcome of each spin.

Type: Virtual Manipulative

This virtual manipulative is intended to introduce users to the idea of visual representation of data by means of a bar chart. This manipulative is also useful for letting the students categorize collections of familiar objects, (shoes, dry food items, coins etc.) separate them into sub-categories, and describe.

Type: Virtual Manipulative

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

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

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

Type: Virtual Manipulative

This manipulative will help students in understanding scatter plots which are particularly useful when investigating whether there is a relationship between two variables. Students could develop a systematic plan for collecting and entering data into the scatter plot manipulative and set appropriate ranges for the *x* and *y* scales.

Type: Virtual Manipulative

In this activity, students adjust slider bars which adjust the coefficients and constants of a linear function and examine how their changes affect the graph. The equation of the line can be in slope-intercept form or standard form. This activity allows students to explore linear equations, slopes, and y-intercepts and their visual representation on a graph. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

In this activity, students adjust how many sections there are on a fair spinner then run simulated trials on that spinner as a way to develop concepts of probability. A table next to the spinner displays the theoretical probability for each color section of the spinner and records the experimental probability from the spinning trials. This activity allows students to explore the topics of experimental and theoretical probability by seeing them displayed side by side for the spinner they have created. 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

With this online Java applet, students use slider bars to move a cross section of a cone, cylinder, prism, or pyramid. This activity allows students to explore conic sections and the 3-dimensional shapes from which they are derived. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

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

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

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

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

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

Type: Virtual Manipulative

This virtual manipulative will help the students in exploring the prime factorization of numbers and see how to use the factorization of a pair of numbers to find the greatest common factor (GCF) and the least common factor (LCM). In the manipulative, the number pairs are presented randomly, so that a student returning to the factor tree will most likely begin with a pair of numbers not seen before.

Type: Virtual Manipulative

This manipulative is a virtual realization of the kind of physical experience that might be available to students given three pieces of straws and told to make them into a triangle. when working with pieces that determine unique triangles (SSS, SAS, ASA). Students construct triangles with the parts provided. After building a red and a blue triangle, students can experience congruence by actually moving one on the top of the other.

Type: Virtual Manipulative

This extremely versatile manipulative can be used by learners of different grades. At early grades, this manipulative will help the students recognize, name, build, draw, and compare two-dimensional shapes. As they progress students can classify and understand relationships among types of two-dimensional objects using their defining properties. The application computes perimeter and area allowing students to explore patterns as dimensions change.

Type: Virtual Manipulative

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

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

In this activity, students use preset data or enter in their own data to be represented in a box plot. This activity allows students to explore single as well as side-by-side box plots of different data. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.

Type: Virtual Manipulative

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

Type: Virtual Manipulative

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

Type: Virtual Manipulative

This online manipulative allows the student to simulate placing marbles into a bag and finding the probability of pulling out certain combinations of marbles. This allows exploration of probabilities of multiple events as well as probability with and without replacement. The tabs above the applet provide access to supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.

Type: Virtual Manipulative

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

Type: Virtual Manipulative

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

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

The students will play a classic game from a popular show. Through this they can explore the probability that the ball will land on each of the numbers and discover that more accurate results coming from repeated testing. The simulation can be adjusted to influence fairness and randomness of the results.

Type: Virtual Manipulative

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

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

Type: Virtual Manipulative

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

Type: Virtual Manipulative

Explore probability topics by modeling coin tossing, free throwing shooting, and manufacturing defects with this virtual manipulative.

Type: Virtual Manipulative

A versatile tool that can be used to illustrate the operations of addition, subtraction, multiplication, and division.

Type: Virtual Manipulative

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

Users select a data set or enter their own data to generate a box plot.

Type: Virtual Manipulative

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

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

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

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

Type: Virtual Manipulative

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

Type: Virtual Manipulative

This virtual manipulative allows one to make a random drawing box, putting up to 21 tickets with the numbers 0-11 on them. After selecting which tickets to put in the box, the applet will choose tickets at random. There is also an option which will show the theoretical probability for each ticket.

Type: Virtual Manipulative

This tool helps students better understand that equality is a relationship and not an operational command to "find the answer." The applet features a pan balance that allows the student to input each half of an equation in the pans, which responds to the numerical expression's value by raising, lowering or balancing.

Type: Virtual Manipulative

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

Type: Virtual Manipulative

Explore the effect on perimeter and area of two rectangular shapes as the scale factor changes.

Type: Virtual Manipulative

Using an interactive applet, students can compare and contrast properties of measures of central tendency, specifically the influence of changes in data values on the mean and median. As students change the data values by dragging the red points to the left or right, the interactive figure dynamically adjusts the mean and median of the new data set.

(NCTM's Illuminations)

Type: Virtual Manipulative

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

In this activity, students practice solving algebraic expressions using order of operations. The applet records their score so the student can track their progress. This activity allows students to practice applying the order of operations when solving problems. 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

In this activity, students can create and view a histogram using existing data sets or original data entered. Students can adjust the interval size using a slider bar, and they can also adjust the other scales on the graph. This activity allows students to explore histograms as a way to represent data as well as the concepts of mean, standard deviation, and scale. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

## Parent Resources

## Educational Games

This addition game uses mixed decimals to the tenths place. This game encourages some logical analysis as well as addition skills. There may be several ways to make the first couple of circles sum to 3, but there is only one way to combine all the given numbers so that every circle sums to 3.

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

This virtual manipulative poses problems requiring the students to position numbers in a diagram, so all numbers in a line add up to a given value.

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

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

This resource is an online glossary to find the meaning of math terms. Students can also use the online glossary to find words that are related to the word typed in the search box. For example: Type in "transversal" and 11 other terms will come up. Click on one of those terms and its meaning is displayed.

Type: Educational Software / Tool

## Image/Photograph

In this lesson, you will find clip art and various illustrations of polygons, circles, ellipses, star polygons, and inscribed shapes.

Type: Image/Photograph

## Perspectives Video: Experts

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

Type: Perspectives Video: Expert

It's impossible to count every animal in a park, but with statistics and some engineering, biologists can come up with a good estimate.

Type: Perspectives Video: Expert

How do scientists collect information from the world? They sample it! Learn how scientists take samples of phytoplankton not only to monitor their populations, but also to make inferences about the rest of the ecosystem!

Type: Perspectives Video: Expert

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

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

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

Type: Perspectives Video: Professional/Enthusiast

Find out how math and technology can help you (try to) get away from civilization.

Type: Perspectives Video: Professional/Enthusiast

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

Type: Perspectives Video: Professional/Enthusiast

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

Sometimes scientists conduct a census, too! Learn how population sampling can help monitor the progress of an ecological restoration project.

Type: Perspectives Video: Professional/Enthusiast

## Problem-Solving Tasks

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.

Type: Problem-Solving Task

In this task students are given a tile pattern involving congruent regular octagons and squares. They are asked to determine the interior angle measure of the octagon and verify the attributes of the square.

Type: Problem-Solving Task

Students are asked to use a diagram or table to write an algebraic expression and use the expression to solve problems.

Type: Problem-Solving Task

This task asks students to calculate probabilities using information presented in a two-way frequency table.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

This tasks gives a verbal description for computing the perimeter of a rectangle and asks the students to find an expression for this perimeter. They then have to use the expression to evaluate the perimeter for specific values of the two variables.

Type: Problem-Solving Task

Students are asked to determine if given expressions are equivalent.

Type: Problem-Solving Task

The purpose of this task is to ask students to write expressions and to consider what it means for two expressions to be equivalent.

Type: Problem-Solving Task

Students are asked to explore and then write an expression with an exponent. The purpose of this task is to introduce the idea of exponential growth and then connect that growth to expressions involving exponents. It illustrates well how fast exponential expressions grow.

Type: Problem-Solving Task

This problem asks the student to find a 3% sales tax on a vase valued at $450.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

The purpose of this task is two-fold. One is to provide students with a multi-step problem involving volume. The other is to give them a chance to discuss the difference between exact calculations and their meaning in a context. It is important to note that students could argue that whether the new pan is appropriate depends in part on how accurate Leo's estimate for the needed height is.

Type: Problem-Solving Task

Students are asked to determine and illustrate all possible descriptions for the base and height of a given triangle.

Type: Problem-Solving Task

Students are asked to draw a scale model of a building and find related volume and surface areas of the model and the building which are rectangular prisms.

Type: Problem-Solving Task

Students are asked to demonstrate two different strategies for finding the area of polygons shown on grids.

Type: Problem-Solving Task

Students are asked to use the given information to determine the cost of painting a barn.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

The purpose of this task is to gain a better understanding of factors and common factors. Students should use the distributive property to show that the sum of two numbers that have a common factor is also a multiple of the common factor.

Type: Problem-Solving Task

The purpose of this task requires students to apply the concepts of factors and common factors in a context. A version of this task could be adapted into a teaching task to help motivate the need for the concept of a common factor.

Type: Problem-Solving Task

Students are asked to reason about and explain the position of two locations relative to sea level.

Type: Problem-Solving Task

The purpose of this task is for students to solve problems involving decimals in a context involving a concept that supports financial literacy, namely inflation. Inflation is a sustained increase in the average price level. In this task, students are asked to compare the buying power of $20 in 1987 and 2012, at least with respect to movie tickets.

Type: Problem-Solving Task

Given the fact 13 x 17 = 221, students are asked to reason about and explain the decimal placement in multiplication and division problems where some of the numbers involved have been changed by powers of ten.

Type: Problem-Solving Task

Students are asked to reason about and explain the placement of decimals in quotients.

Type: Problem-Solving Task

Students are asked to solve a distance problem involving fractions.

Type: Problem-Solving Task

Students are asked to solve a fraction division problem using both a visual model and the standard algorithm within a real-world context.

Type: Problem-Solving Task

The purpose of this task is converting square units. Use the information provided to answer the questions posed. Since this task asks students to critique Jada's reasoning, it provides an opportunity to work on Standard for Mathematical Practice MAFS.K12.MP.3.1 - Construct Viable Arguments and Critique the Reasoning of Others.

Type: Problem-Solving Task

Students are asked to use a ratio to determine how much money Jim and Jesse had at the start of their trip.

Type: Problem-Solving Task

This task does address some aspects of modeling as described in Florida Standard for Mathematical Practice 4. Also, students often think that time must always be the independent variable and so may need some help understanding that one chooses the independent and dependent variable based on the way one wants to view a situation.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Students are asked to determine the percent of the area of a store covered by a security camera. Then, students are asked to determine the "best" place to position the camera and support their answer.

Type: Problem-Solving Task

Use the information provided to find out the original price of Selina's shirt. There are several different ways to reason through this problem; two approaches are shown.

Type: Problem-Solving Task

This problem is the fifth in a series of seven about ratios. At first glance the problem may look to be beyond MAFS.6.RP.1.3, which limits itself to "describe a ratio relationship between two quantities." However, even though there are three quantities (the number of each candidates' votes), they are only considered two at a time.

Type: Problem-Solving Task

This is the sixth problem in a series of seven that use the context of a classroom election. While it still deals with simple ratios and easily managed numbers, the mathematics surrounding the ratios are increasingly complex. In this problem, the students are asked to determine the difference in votes received by two of the three candidates.

Type: Problem-Solving Task

This is the last problem of seven in a series about ratios set in the context of a classroom election. Since the number of voters is not known, the problem is quite abstract and requires a deep understanding of ratios and their relationship to fractions.

Type: Problem-Solving Task

This problem is the third in a series of tasks set in the context of a class election. Students are given a ratio and total number of voters and are asked to determine the difference between the winning number of votes received and the number of votes needed for victory.

Type: Problem-Solving Task

This is the first and most basic problem in a series of seven problems, all set in the context of a classroom election. Students are given a ratio and total number of voters and are asked to determine the number of votes received by each candidate.

Type: Problem-Solving Task

This is the second in a series of tasks that are set in the context of a classroom election. It requires students to understand what ratios are and apply them in a context. The simple version of this question just asked how many votes each gets. This has the extra step of asking for the difference between the votes.

Type: Problem-Solving Task

This is the fourth in a series of tasks about ratios set in the context of a classroom election. Given only a ratio, students are asked to determine the fractional difference between votes received and votes required.

Type: Problem-Solving Task

Students are given a context and a dotplot and are asked a number of questions regarding shape, center, and spread of the data.

Type: Problem-Solving Task

Students are given a context and a series of questions and are asked to identify whether each question is statistical and to provide their reasoning. Students are asked to compose an original statistical question for the given context.

Type: Problem-Solving Task

Using the information provided, create an appropriate graphical display and answer the questions regarding shape, center and variability.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

The purpose of this task is for students to translate between measurements given in a scale drawing and the corresponding measurements of the object represented by the scale drawing. If used in an instructional setting, it would be good for students to have an opportunity to see other solution methods, perhaps by having students with different approaches explain their strategies to the class. Students who can only solve this by first converting the linear measurements will have a hard time solving problems where only area measures are given.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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?

Type: Problem-Solving Task

Students must use the information to answer the multiple tasks provided.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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?

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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?

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

In this problem students are comparing a very small quantity with a very large quantity using the metric system. The metric system is especially convenient when comparing measurements using scientific notations since different units within the system are related by powers of ten.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

In this activity, the student is asked to solve a variety of equations (one solution, infinite solutions, no solution) in the traditional algebraic manner and to use pictures of a pan balance to show the solution process.

Type: Problem-Solving Task

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).

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

It is possible to say a lot about the solution to an equation without actually solving it, just by looking at the structure and operations that make up the equation. This exercise turns the focus away from the familiar "finding the solution" problem to thinking about what it really means for a number to be a solution of an equation.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

This task could be put to good use in an instructional sequence designed to develop knowledge related to students' understanding of linear functions in contexts. Though students could work independently on the task, collaboration with peers is more likely to result in the exploration of a range of interpretations.

Type: Problem-Solving Task

This task has students engaging in a simple modeling exercise, taking verbal and numerical descriptions of battery life as a function of time and writing down linear models for these quantities. To draw conclusions about the quantities, students have to find a common way of describing them.

Type: Problem-Solving Task

The purpose of this task is for students to interpret two distance-time graphs in terms of the context of a bicycle race. There are two major mathematical aspects to this: interpreting what a particular point on the graph means in terms of the context and understanding that the "steepness" of the graph tells us something about how fast the bicyclists are moving.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

This task lets students explore the differences between linear and non-linear functions. By contrasting the two, it reinforces properties of linear functions.

Type: Problem-Solving Task

The primary purpose of this task is to elicit common misconceptions that arise when students try to model situations with linear functions. This task, being multiple choice, could also serve as a quick assessment to gauge a class' understanding of modeling with linear functions.

Type: Problem-Solving Task

This is a simple task about interpreting the graph of a function in terms of the relationship between quantities that it represents.

Type: Problem-Solving Task

In this task students draw the graphs of two functions from verbal descriptions. Both functions describe the same situation but changing the viewpoint of the observer changes where the function has output value zero. This small twist forces the students to think carefully about the interpretation of the dependent variable. This task could be used in different ways: To generate a class discussion about graphing. As a quick assessment about graphing, for example during a class warm-up. To engage students in small group discussion.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Students will just be learning about similarity in this grade, so they may not recognize that it is needed in this context. Teachers should be prepared to give support to students who are struggling with this part of the task. To simplify the task, the teacher can just tell the students that based on the slant of the truncated conical cup, the complete cone would be 14 in tall and the part that was sliced off was 10 inches tall. (See solution for an explanation.) There is a worthwhile discussion to be had about parts (c) and (e). The percentage increase is smaller for the snow cones than it was for the juice treats. The snow cones have volume which is equal to those of the juice treats plus the volume of the dome, which is the same in both cases. Adding the same number to two numbers in a ratio will always make their ratio closer to one, which in this case means that the ratio - and thus percentage increase - would be smaller.

Type: Problem-Solving Task

Students' first experience with transformations is likely to be with specific shapes like triangles, quadrilaterals, circles, and figures with symmetry. Exhibiting a sequence of transformations that shows that two generic line segments of the same length are congruent is a good way for students to begin thinking about transformations in greater generality.

Type: Problem-Solving Task

This task has two goals: first to develop student understanding of rigid motions in the context of demonstrating congruence. Secondly, student knowledge of reflections is refined by considering the notion of orientation in part (b). Each time the plane is reflected about a line, this reverses the notions of ''clockwise'' and ''counterclockwise.''

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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).

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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?

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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

Type: Problem-Solving Task

In a poll of Mr. Briggs's math class, 67% of the students say that math is their favorite academic subject. The editor of the school paper is in the class, and he wants to write an article for the paper saying that math is the most popular subject at the school. Explain why this is not a valid conclusion and suggest a way to gather better data to determine what subject is most popular.

Type: Problem-Solving Task

In this task, students are able to conjecture about the differences and similarities in the two groups from a strictly visual perspective and then support their comparisons with appropriate measures of center and variability. This will reinforce that much can be gleaned simply from visual comparison of appropriate graphs, particularly those of similar scale.

Type: Problem-Solving Task

The purpose of this task is to provide students with the opportunity to determine experimental probabilities by collecting data. The cylindrical objects used in this task typically have three different resting positions but not all of these may be equally likely and some may be extremely unlikely or impossible when the object is tossed. Furthermore, obtaining the probabilities of the outcomes is perhaps only possible through the use of long-run relative frequencies. This is because these cylinders do not have the same types of symmetries as objects that are often used as dice, such as cubes or tetrahedrons, where each outcome is equally likely.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

This resource involves a simple data-gathering activity which furnishes data that students organize into a table. They are then asked to refer to the data and determine the probability of various outcomes.

Type: Problem-Solving Task

This task introduces the fundamental statistical ideas of using data summaries (statistics) from random samples to draw inferences (reasoned conclusions) about population characteristics (parameters). In the task built around an election poll scenario, the population is the entire seventh grade class, the unknown characteristic (parameter) of interest is the proportion of the class members voting for a specific candidate, and the sample summary (statistic) is the observed proportion of voters favoring the candidate in a random sample of class members. Variation 2 leads students through a physical simulation for generating sample proportions by sampling, and re-sampling, marbles from a box.

Type: Problem-Solving Task

This task introduces the fundamental statistical ideas of using data summaries (statistics) from random samples to draw inferences (reasoned conclusions) about population characteristics (parameters). There are two important goals in this task: seeing the need for random sampling and using randomization to investigate the behavior of a sample statistic. These introduce the basic ideas of statistical inference and can be accomplished with minimal knowledge of probability.

Type: Problem-Solving Task

As the standards in statistics and probability unfold, students will not yet know the rules of probability for compound events. Thus, simulation is used to find an approximate answer to these questions. In fact, part b would be a challenge to students who do know the rules of probability, further illustrating the power of simulation to provide relatively easy approximate answers to wide-ranging problems.

Type: Problem-Solving Task

This task is intended as a classroom activity. Students pool the results of many repetitions of the random phenomenon (rolling dice) and compare their results to the theoretical expectation they develop by considering all possible outcomes of rolling two dice. This gives them a concrete example of what we mean by long term relative frequency.

Type: Problem-Solving Task

The purpose of this task is for students to compute the theoretical probability of a compound event. Teachers may wish to emphasize the distinction between theoretical and experimental probabilities for this problem. For students learning to distinguish between theoretical and experimental probability, it would be good to find an experimental probability either before or after students have calculated the theoretical probability.

Type: Problem-Solving Task

The purpose of this task is for students to compute the theoretical probability of a seating configuration. There are 24 possible configurations of the four friends at the table in this problem. Students could draw all 24 configurations to solve the problem but this is time consuming and so they should be encouraged to look for a more systematic method.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

Type: Problem-Solving Task

This task provides us with the opportunity to see how the mathematical ideas embedded in the standards and clusters mature over time. The task "Uses facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure (MAFS.7.G.2.5)" except that it requires students to know, in addition, something about parallel lines, which students will not see until 8th grade. As a result, this task is especially good at illustrating the links between related standards across grade levels.

Type: Problem-Solving Task

The purpose of this task is to give students practice working the formulas for the volume of cylinders, cones and spheres, in an engaging context that provides and opportunity to attach meaning to the answers.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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."

Type: Problem-Solving Task

When students plot irrational numbers on the number line, it helps reinforce the idea that they fit into a number system that includes the more familiar integer and rational numbers. This is a good time for teachers to start using the term "real number line" to emphasize the fact that the number system represented by the number line is the real numbers. When students begin to study complex numbers in high school, they will encounter numbers that are not on the real number line (and are, in fact, on a "number plane"). This task could be used for assessment, or if elaborated a bit, could be used in an instructional setting.

Type: Problem-Solving Task

Students should think of different ways the cylindrical containers can be set up in a rectangular box. Through the process, students should realize that although some setups may seem different, they result in a box with the same volume. In addition, students should come to the realization (through discussion and/or questioning) that the thickness of a cardboard box is very thin and will have a negligible effect on the calculations.

Type: Problem-Solving Task

This task is ideally suited for instruction purposes where students can take their time and develop several of the Mathematical Practice standards, as the mathematical content is directly related to, but somewhat exceeds, the content of standard MAFS.8.G.1.5 on sums of angles in triangles. Careful analysis of the angles requires students to construct valid arguments (MAFS.K12.MP.3.1) using abstract and quantitative reasoning (MAFS.K12.MP.2.1). Producing the picture in part (c) helps students identify a common mathematical argument repeated multiple times (MAFS.K12.MP.8.1). If students use pattern blocks in order to develop the intuition for decomposing the hexagon into triangles, then this is also an example of MAFS.K12.MP.5.1.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Students are given a pair of numbers. They are asked to determine which is larger, and then justify their answer. The numbers involved are rational numbers and common irrational numbers, such π and square roots. This task can be used to either build or assess initial understandings related to rational approximations of irrational numbers.

Type: Problem-Solving Task

This task asks the student to gather data on whether classmates play an instrument and/or participate in a sport, summarize the data in a table and decide whether there is an association between playing a sport and playing an instrument. Finally, the student is asked to create a graph to display any association between the variables.

Type: Problem-Solving Task

Students are asked to examine data given in table format and then calculate either row percentages or column percentages and state a conclusion about the meaning of the data. Either calculation is appropriate for the solution since there is no clear relationship between the variables. Whether the student sees a strong association or not is less important than whether his or her answer uses the data appropriately and demonstrates understanding that an association means the distribution of favorite subject is different for 7th graders and 8th graders.

Type: Problem-Solving Task

Students are asked to examine a scatter plot and then interpret its meaning. Students should identify the form of the relationship (linear, curved, etc.), the direction or correlation (positive or negative), any specific outliers, the strength of the relationship between the two variables, and any other relevant observations.

Type: Problem-Solving Task

In this resource, real-world bivariate data is displayed in a scatter plot. The equation of the linear function which models the relationship between the two variables is provided, and it is graphed on the scatter plot. Students are asked to use the model to interpret the data and to make predictions.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

In this task students interpret two graphs that look the same but show very different quantities. The first graph gives information about how fast a car is moving while the second graph gives information about the position of the car. This problem works well to generate a class or small group discussion. Students learn that graphs tell stories and have to be interpreted by carefully thinking about the quantities shown.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Students are asked a series of questions involving a fraction and a whole number within the context of a recipe. Students are asked to solve a problem using both a visual model and the standard algorithm.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Students are asked to solve a distance problem involving fractions. The purpose of this task is to help students extend their understanding of division of whole numbers to division of fractions, and given the simple numbers used, it is most appropriate for students just learning about fraction division because it lends itself easily to a pictorial solution.

Type: Problem-Solving Task

The purpose of this task is for students to solve problems involving multiplication and division of decimals in the real-world context of setting financial goals. The focus of the task is on modeling and understanding the concept of setting financial goals, so fluency with the computations will allow students to focus on other aspects of the task.

Type: Problem-Solving Task

In this problem-solving task students are challenged to apply their understanding of linear relationships to determine the amount of chicken and steak needed for a barbecue, which will include creating an equation, sketching a graph, and interpreting both. This resource also includes standards alignment commentary and annotated solutions.

Type: Problem-Solving Task

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).

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Students are asked to solve a real-world problem involving common multiples.

Type: Problem-Solving Task

Students are asked to use fractions to determine how many hours it will take a car to travel a given distance.

Type: Problem-Solving Task

Students are asked to use fractions to determine how long a video game can be played.

Type: Problem-Solving Task

The purpose of this task is to have students convert multiple currencies to answer the problem. Students may find the CDN abbreviation for Canada confusing. Teachers may need to explain the fact that money in Canada is also called dollars, so to distinguish them, we call them Canadian dollars.

Type: Problem-Solving Task

Use the information provided to find out what percentage of Dana's lot won't be covered by the house.

Type: Problem-Solving Task

This task asks the students to solve a real-world problem involving unit rates (data per unit time) using units that many teens and pre-teens have heard of but may not know the definition for. While the computations involved are not particularly complex, the units will be abstract for many students. The first solution relies more on reasoning about the meaning of multiplication and division, while the second solution uses units to help keep track of the steps in the solution process.

Type: Problem-Solving Task

Type: Problem-Solving Task

This problem-solving task asks students to find a linear function that models something in the real world. After finding the equation of the linear relationship between the depth of the water and the distance across the channel, students have to verbalize the meaning of the slope and intercept of the line in the context of this situation. Commentary on standards alignment and illustrated solutions are also included.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

The student is asked to perform operations with numbers expressed in scientific notation to decide whether 7% of Americans really do eat at Giantburger every day.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

Students are asked to use knowledge of rates and ratios to answer a series of questions involving time, distance, and speed.

Type: Problem-Solving Task

Students are asked to write complete sentences to describe ratios for the context.

Type: Problem-Solving Task

The purpose of this task is to help students get a better understanding of multiplying and dividing using fractions.

Type: Problem-Solving Task

There are two aspects to fluency with division of multi-digit numbers: knowing when it should be applied, and knowing how to compute it. While this task is very straightforward, it represents the kind of problem that sixth graders should be able to recognize and solve relatively quickly. Easily recognizing contexts that require division is a necessary conceptual prerequisite to more complex modeling problems that students will be asked to solve later in middle and high school.

This task also has a natural carryover to work with ratios and rates, so students should also be building connections between these kinds of division problems and finding unit rates.

Type: Problem-Solving Task

The purpose of the task is for students to compare signed numbers in a real-world context.

Type: Problem-Solving Task

The purpose of this task is to help students explore the meaning of fraction division and to connect it to what they know about whole-number division. Students are asked to explain why the quotient of two fractions with common denominators is equal to the quotient of the numerators of those fractions.

Type: Problem-Solving Task

This task builds on a fifth grade fraction multiplication task, "Drinking Juice." This task uses the identical context, but asks the corresponding "Number of Groups Unknown" division problem. See "Drinking Juice, Variation 3" for the "Group Size Unknown" version.

Type: Problem-Solving Task

Students are asked to solve a fraction division problem using a visual model and the standard algorithm.

Type: Problem-Solving Task

Students are asked to solve problems from context by using multiplication or division of decimals.

Type: Problem-Solving Task

This instructional task requires that the students model each problem with some type of fractions manipulatives or drawings. This could be pattern blocks, student or teacher-made fraction strips, or commercially produced fraction pieces. At a minimum, students should draw pictures of each. The above problems are meant to be a progression which require more sophisticated understandings of the meaning of fractions as students progress through them.

Type: Problem-Solving Task

The purpose of this task is for students to get a better understanding of the relative positions and values of positive and negative numbers.

Type: Problem-Solving Task

The purpose of this task is for students to apply their knowledge of integers in a real-world context.

Type: Problem-Solving Task

Students are asked to add or subtract decimals to solve problems in context.

Type: Problem-Solving Task

Students are asked to write and solve an equation in one variable to answer a real world question.

Type: Problem-Solving Task

In this task students use different representations to analyze the relationship between two quantities and to solve a real world problem. The situation presented provides a good opportunity to make connections between the information provided by tables, graphs and equations. In the later part of the problem, the numbers are big enough so that using the formula is the most efficient way to solve the problem; however, creative use of the table or graph will also work.

Type: Problem-Solving Task

This task asks students to find equivalent expressions by visualizing a familiar activity involving distance. The given solution shows some possible equivalent expressions, but there are many variations possible.

Type: Problem-Solving Task

Students are asked to use properties of operations to match expressions that are equivalent and to write equivalent expressions for any expressions that do not have a match.

Type: Problem-Solving Task

In this task students are asked to write an equation to solve a real-world problem.

Type: Problem-Solving Task

Students are asked to write and graph two inequalities described in context: one discrete and one continuous.

Type: Problem-Solving Task

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

Type: Problem-Solving Task

Students are asked to write an equation with one variable in order to find the distance walked.

Type: Problem-Solving Task

This purpose of this task is to help students understand the absolute value of a number as its distance from 0 on the number line. The context is not realistic, nor is meant to be; it is a thought experiment to help students focus on the relative position of numbers on the number line.

Type: Problem-Solving Task

Students are asked to determine if two different ratios are both appropriate for the same context.

Type: Problem-Solving Task

Given a ratio, students are asked to determine how much of each ingredient is needed to make concrete.

Type: Problem-Solving Task

This problem provides an interesting geometric context to work on the notion of percent. Two different methods for analyzing the geometry are provided: the first places the two squares next to one another and then moves one so that they overlap. The second solution sets up an equation to find the overlap in terms of given information which reflects the mathematical ideas described in cluster MAFS.6.EE.2 - Reason about and solve one-variable equations and inequalities.

Type: Problem-Solving Task

Students are asked to use a given ratio to determine if two different interpretations of the ratio are correct and to determine the maximum quantity that could be purchased within a given context.

Type: Problem-Solving Task

Students are asked apply knowledge of ratios to answer several questions regarding speed, distance and time.

Type: Problem-Solving Task

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.

Type: Problem-Solving Task

In this activity students calculate the ratio of chocolate to cereal when making a cake. Students then use that ratio to calculate to amount of chocolate and cereal necessary to make 21 cakes.

Type: Problem-Solving Task

## Teaching Ideas

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

In this hands-on and web interactive project, students design and build a bird wing powerful enough to spin them in an office chair when it is flapped. By modifying the shape, size, and/or materials used in their design based on observations of natural and man-made transportation methods, students will learn about thrust, forces, durability, and energy use.

Type: Teaching Idea

In this hands-on and web interactive project, students design and build a machine inspired by animals where the entire structure flips or jumps (vertically or horizontally) using basic materials such as sticks and rubber bands. The students will explore concepts including power amplification, elastic potential energy, and kinetic energy by manipulating physical objects.

Type: Teaching Idea

In this task, students start by examining a scatter plot about the size of various bird eggs from a collection of measurements. In particular, students are asked to identify a correlation, sketch an approximation for the line of best fit, determine the equation of that line, use the equation of the line and/or the graph to make interpolative predictions, and draw conclusions about the properties of specific eggs by using the graphical presentation of the data.

Type: Teaching Idea

This resource features two pairs of interactive graphs to help students explore rate of change and linear relationships. "Users can drag a slider on an interactive graph to modify a rate of change (cost per minute for phone use) and learn how modifications in that rate affect the linear graph displaying accumulation (the total cost of calls). In this first part, Constant Cost per Minute, the cost per minute for phone use remains constant over time. In the second part, Changing Cost per Minute, the cost per minute for phone use changes after the first sixty minutes of calls." (from NCTM's Illuminations)

Type: Teaching Idea

## Tutorials

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

Type: Tutorial

The distributive property states that the terms of addition or subtraction statements within parentheses may be separately multiplied by a value outside the parentheses. In this tutorial, students will learn the distributive property, which is very helpful with mental math calculations and solving equations.

Type: Tutorial

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

Type: Tutorial

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

Type: Tutorial

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

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

Type: Tutorial

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

Type: Tutorial

The video demonstrates rewriting given numbers in a common format (as decimals), so they can be compared and ordered.

Type: Tutorial

Introduction to solving one variable multiplication equations of the form px = q.

Type: Tutorial

## Video/Audio/Animations

Ratio errors confuse one of the coaches as two teams face off in an epic dodgeball tournament. See how mathematical techniques such as tables, graphs, measurements and equations help to find the missing part of a proportion.

Atlantean Dodgeball addresses number and operations standards, the algebra standard, and the process standard, as established by the National Council of Teachers of Mathematics (NCTM). It guides students in:

- Understanding and using ratios and proportions to represent quantitative relationships.
- Relating and comparing different forms of representation for a relationship.
- Developing, analyzing, and explaining methods for solving problems involving proportions, such as scaling and finding equivalent ratios.
- Representing, analyzing, and generalizing a variety of patterns with tables, graphs, words, and, when possible, symbolic rules.

Type: Video/Audio/Animation

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

Type: Video/Audio/Animation

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

Type: Video/Audio/Animation

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

Type: Video/Audio/Animation

## Virtual Manipulatives

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

This virtual manipulative is a probability simulation tool which will help the learners to understand the concepts of experimental and theoretical probability.

Type: Virtual Manipulative

The purpose of this manipulative is to help students recognize that (1) unusual events do happen, and (2) it may take a longer time for some of them to happen. The letters are drawn at random from the beginning of Hamlet's soliloquy, "To be, or not to be." Any word made from those letters (such as TO) can be entered in the box. When the *start* is pressed, letters are drawn and recorded. The process continues until the word appears.

Type: Virtual Manipulative

This drag and drop Venn diagram simulation gives students the opportunity to solve a mathematical problem based on number properties using a range of different Venn diagrams. There are five different levels involving a range of multiples and simply odds and evens. The three core layouts cover simple separate sets, two intersecting sets, and a three way intersecting Venn Diagram. The odds and evens layout is limited to two intersecting sets, of course.

Type: Virtual Manipulative

This virtual manipulative can be used demonstrate random probability and to teach about chance and random choices. Use this free, fully customizable, online spinner to create probability scenarios involving numerous choices, or create advanced, unevenly split spinners to demonstrate and model real life scenarios.

This spinner also incorporates a bar graph to record and model the outcome of each spin.

Type: Virtual Manipulative

This virtual manipulative is intended to introduce users to the idea of visual representation of data by means of a bar chart. This manipulative is also useful for letting the students categorize collections of familiar objects, (shoes, dry food items, coins etc.) separate them into sub-categories, and describe.

Type: Virtual Manipulative

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

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

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

Type: Virtual Manipulative

This manipulative will help students in understanding scatter plots which are particularly useful when investigating whether there is a relationship between two variables. Students could develop a systematic plan for collecting and entering data into the scatter plot manipulative and set appropriate ranges for the *x* and *y* scales.

Type: Virtual Manipulative

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

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

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

Type: Virtual Manipulative

This virtual manipulative will help the students in exploring the prime factorization of numbers and see how to use the factorization of a pair of numbers to find the greatest common factor (GCF) and the least common factor (LCM). In the manipulative, the number pairs are presented randomly, so that a student returning to the factor tree will most likely begin with a pair of numbers not seen before.

Type: Virtual Manipulative

This manipulative is a virtual realization of the kind of physical experience that might be available to students given three pieces of straws and told to make them into a triangle. when working with pieces that determine unique triangles (SSS, SAS, ASA). Students construct triangles with the parts provided. After building a red and a blue triangle, students can experience congruence by actually moving one on the top of the other.

Type: Virtual Manipulative

This extremely versatile manipulative can be used by learners of different grades. At early grades, this manipulative will help the students recognize, name, build, draw, and compare two-dimensional shapes. As they progress students can classify and understand relationships among types of two-dimensional objects using their defining properties. The application computes perimeter and area allowing students to explore patterns as dimensions change.

Type: Virtual Manipulative

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

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

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

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

Type: Virtual Manipulative

Explore probability topics by modeling coin tossing, free throwing shooting, and manufacturing defects with this virtual manipulative.

Type: Virtual Manipulative

A versatile tool that can be used to illustrate the operations of addition, subtraction, multiplication, and division.

Type: Virtual Manipulative

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

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

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

Type: Virtual Manipulative

Section:Grades PreK to 12 Education Courses >Grade Group:Grades 6 to 8 Education Courses >Subject:Mathematics >SubSubject:Remedial Mathematics >