Course Standards
General Course Information and Notes
General Notes
Additional content addressed on the Grade 8 NAEP Mathematics assessment includes:
 Draw or sketch from a written description polygons, circles, or semicircles. (MAFS.7.G.1.2; include circles and semicircles)
 Represent or describe a threedimensional situation in a twodimensional drawing from different views. (MAFS.6.G.1.4)
 Demonstrate an understanding about the two and threedimensional shapes in our world through identifying, drawing, modeling, building, or taking apart. (MAFS.6.G.1.4, MAFS.7.G.1.3, MAFS.7.G.2.6)
 Visualize or describe the cross section of a solid. (MAFS.7.G.1.3)
 Represent geometric figures using rectangular coordinates on a plane. (MAFS.6.G.1.3)
 Describe how mean, median, mode, range, or interquartile ranges relate to distribution shape. (MAFS.6.SP.2.5c)
 Using appropriate statistical measures, compare two or more data sets describing the same characteristic for two different populations for subset of the same population. (MAFS.7.SP.2.3, MAFS.7.SP.2.4)
 Given a sample, identify possible sources of bias in sampling. (MAFS.7.SP.1.1)
 Distinguish between a random and nonrandom sample. (MAFS.7.SP.1.1)
 Evaluate the design of an experiment. (MAFS.7.SP.1.2)

Determine the theoretical probability of simple and compound events in familiar contexts. (MAFS.7.SP.3.8a)
 Estimate the probability of simple and compound events through experimentation or simulation. (MAFS.7.SP.3.8)
 Use theoretical probability to evaluate or predict experimental outcomes. (MAFS.7.SP.3.6, MAFS.SP.3.7)
 Describe relative positions of points and lines using the geometric ideas of midpoint, points on common line through a common point, parallelism, or perpendicularity.
 Describe the intersection of two or more geometric figures in the plane (e.g., intersection of a circle and a line).
 Make and test a geometric conjecture about regular polygons.
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
Florida Standards Implementation Guide Focus Section:
The Mathematics Florida Standards Implementation Guide was created to support the teaching and learning of the Mathematics Florida Standards. The guide is compartmentalized into three components: focus, coherence, and rigor.Focus means narrowing the scope of content in each grade or course, so students achieve higher levels of understanding and experience math concepts more deeply. The Mathematics standards allow for the teaching and learning of mathematical concepts focused around major clusters at each grade level, enhanced by supporting and additional clusters. The major, supporting and additional clusters are identified, in relation to each grade or course. The cluster designations for this course are below.
Major Clusters
MAFS.8.EE.1 Work with radicals and integer exponents.
MAFS.8.EE.2 Understand the connections between proportional relationships, lines, and linear equations.
MAFS.8.EE.3 Analyze and solve linear equations and pairs of simultaneous linear equations.
MAFS.8.F.1 Define, evaluate, and compare functions.
MAFS.8.F.2 Use functions to model relationships between quantities.
MAFS.8.G.1 Understand congruence and similarity using physical models, transparencies, or geometry software.
MAFS.8.G.2 Understand and apply the Pythagorean Theorem.
Supporting Clusters
MAFS.8.NS.1 Know that there are numbers that are not rational, and approximate them by rational numbers.
MAFS.8.SP.1 Investigate patterns of association in bivariate data.
Additional Clusters
MAFS.G.3 Solve realworld and mathematical problems involving volume of cylinders, cones, and spheres.
Note: Clusters should not be sorted from major to supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting and additional clusters.
Version Requirements
In Grade 8, instructional time should focus on three critical areas: (1) formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations; (2) grasping the concept of a function and using functions to describe quantitative relationships; (3) analyzing two and threedimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem.
 Students use linear equations and systems of linear equations to represent, analyze, and solve a variety of problems. Students recognize equations for proportions (y/x = m or y = mx) as special linear equations (y = mx + b), understanding that the constant of proportionality (m) is the slope, and the graphs are lines through the origin. They understand that the slope (m) of a line is a constant rate of change, so that if the input or xcoordinate changes by an amount A, the output or ycoordinate changes by the amount m(A). Students also use a linear equation to describe the association between two quantities in bivariate data (such as arm span vs. height for students in a classroom). At this grade, fitting the model, and assessing its fit to the data are done informally. Interpreting the model in the context of the data requires students to express a relationship between the two quantities in question and to interpret components of the relationship (such as slope and yintercept) in terms of the situation.
Students strategically choose and efficiently implement procedures to solve linear equations in one variable, understanding that when they use the properties of equality and concept of logical equivalence, they maintain the solutions of the original equation. Students solve systems of two linear equations in two variables and relate the systems to pairs of lines in the plane; these intersect, are parallel, or are the same line. Students use linear equations, systems of linear equations, linear functions, and their understanding of slope of a line to analyze situations and solve problems.  Students grasp the concept of a function as a rule that assigns to each input exactly one output. They understand that functions describe situations where one quantity determines another. They can translate among representations and partial representations of functions (noting that tabular and graphical representations may be partial representations), and they describe how aspects of the function are reflected in the different representations.
 Students use ideas about distance and angles, how they behave under translations, rotations, reflections, and dilation, and ideas about congruence and similarity to describe and analyze twodimensional figures and to solve problems. Students show that the sum of the angles in a triangle is the angle formed by a straight line, and that various configurations of lines give rise to similar triangles because of the angles created when a traversal cuts parallel lines. Students understand the statement of the Pythagorean Theorem and its converse, and can explain why the Pythagorean Theorem holds, for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points on the coordinate plane, to find lengths, and to analyze polygons. Students complete their work on volume by solving problems involving cones, cylinders, and spheres.
Additional Instructional Resources:
A.V.E. for Success Collection: http://www.fasa.net/iTunesU/index.cfm
General Information
 Class Size Core Required
Educator Certifications
Student Resources
Original Student Tutorials
Learn how similar right triangles can show how the slope is the same between any two distinct points on a nonvertical line as you help Hailey build stairs to her tree house in this interactive tutorial.
Type: Original Student Tutorial
Learn how math models can show why social distancing during a epidemic or pandemic is important in this interactive tutorial.
Type: Original Student Tutorial
Learn to construct a function to model a linear relationship between two quantities and determine the slope and yintercept given two points that represent the function with 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 multistep 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 multistep equations in this interactive tutorial.
This is part five of five in a series on solving multistep 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 multistep 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 multistep 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 multistep equations using the distributive property in this interactive tutorial.
This is part two of five in a series on solving multistep 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 multistep equations that contain like terms in this interactive tutorial.
This is part one of five in a series on solving multistep 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
See how sweet it can be to determine the slope of linear functions and compare them in this interactive tutorial. Determine and compare the slopes or the rates of change by using verbal descriptions, tables of values, equations and graphical forms.
Type: Original Student Tutorial
Have some fun with FUNctions! Learn how to identify linear and nonlinear 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
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 inquirybased, handson 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
Continue an exploration of kinematics to describe linear motion by focusing on positiontime measurements from the motion trial in part 1. In this interactive tutorial, you'll identify position measurements from the spark tape, analyze a scatterplot of the positiontime data, calculate and interpret slope on the positiontime graph, and make inferences about the dune buggy’s average speed
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 yintercept 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 6part 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 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 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
Investigate the limiting factors of a Florida ecosystem and describe how these limiting factors affect one native populationthe Florida ScrubJaywith this interactive tutorial.
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 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
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
Play this interactive game and determine whether the similar shapes have gone through rotations, translations, or reflections.
Type: Educational Game
In this timed activity, students solve linear equations (one and twostep) 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: onestep, twostep, or quadratic equations and using the distributive property if desired. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.
Type: Educational Game
Educational Software / Tools
This 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
Perspectives Video: Expert
Don't be a square! Learn about how even grids help archaeologists track provenience!
Download the CPALMS Perspectives video student note taking guide.
Type: Perspectives Video: Expert
Perspectives Video: Professional/Enthusiast
Find out how math and technology can help you (try to) get away from civilization.
Download the CPALMS Perspectives video student note taking guide.
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 Midcontinent Research for Education and Learning: 1) Analyze characteristics and properties of two and threedimensional 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 realworld 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
ProblemSolving Tasks
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: ProblemSolving 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: ProblemSolving 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: ProblemSolving 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: ProblemSolving 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: ProblemSolving 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: ProblemSolving 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: ProblemSolving Task
In this task, students are presented with a realworld 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: ProblemSolving 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: ProblemSolving 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: ProblemSolving Task
This task presents a realworld 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: ProblemSolving 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: ProblemSolving 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: ProblemSolving 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: ProblemSolving 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: ProblemSolving Task
The purpose of this task is for students to interpret two distancetime 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: ProblemSolving 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 inputoutput 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: ProblemSolving 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 noninteger 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: ProblemSolving Task
This task lets students explore the differences between linear and nonlinear functions. By contrasting the two, it reinforces properties of linear functions.
Type: ProblemSolving 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: ProblemSolving Task
This is a simple task about interpreting the graph of a function in terms of the relationship between quantities that it represents.
Type: ProblemSolving 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 warmup. To engage students in small group discussion.
Type: ProblemSolving 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: ProblemSolving 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: ProblemSolving 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: ProblemSolving 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: ProblemSolving Task
In this resource, students experiment with successive reflections of a triangle in a coordinate plane.
Type: ProblemSolving 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 nonsquare integer. Students may need guidance in thinking about how to approach the task.
Type: ProblemSolving 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 xy 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: ProblemSolving 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: ProblemSolving 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: ProblemSolving Task
This task would be especially wellsuited for instructional purposes. Students will benefit from a class discussion about the slope, yintercept, xintercept, and implications of the restricted domain for interpreting more precisely what the equation is modeling.
Type: ProblemSolving 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 angleangle criterion for similarity of triangles.
Type: ProblemSolving 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 multistep 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: ProblemSolving 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: ProblemSolving 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: ProblemSolving 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: ProblemSolving 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: ProblemSolving 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: ProblemSolving 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: ProblemSolving 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: ProblemSolving 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: ProblemSolving 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: ProblemSolving 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: ProblemSolving 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: ProblemSolving Task
In this resource, realworld 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: ProblemSolving 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: ProblemSolving 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: ProblemSolving 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: ProblemSolving 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 shortterm capital for investment or reinvestment in developing the business.
Type: ProblemSolving Task
In this problemsolving 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: ProblemSolving 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: ProblemSolving 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: ProblemSolving 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: ProblemSolving 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: ProblemSolving Task
In this problemsolving 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: ProblemSolving Task
This problemsolving 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: ProblemSolving Task
This task asks the student to understand the relationship between slope and changes in x and yvalues of a linear function.
Type: ProblemSolving Task
This activity challenges students to recognize the relationship between slope and the difference in x and yvalues 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 yintercepts. This task has also produced a reasonable starting place for discussing pointslope form of a linear equation.
Type: ProblemSolving 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: ProblemSolving 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: ProblemSolving 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: ProblemSolving Task
The student is given the equation 5x2y=3 and asked, if possible, to write a second linear equation creating systems resulting in one, two, infinite, and no solutions.
Type: ProblemSolving Task
In this problemsolving 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: ProblemSolving Task
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 video, we find missing angle measures from a variety of examples.
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
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 nonlinear 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 slopeintercept 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 yintercept (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 realworld examples of linear graphs and interpret the relationship between the two variables.
Type: Tutorial
In this video, you will practice writing the slopeintercept 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 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 yintercept. You will then write the equation of the line in slopeintercept form.
Type: Tutorial
Students will learn how to graph a linear equation using a table. Students will not be required to graph from slopeintercept form, although they will convert the equation from standard form to slopeintercpet 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 slopeintercept form.
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
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 slopeintercept form from graphs. You will then practice graphing lines from equations in slopeintercept 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 slopeintercept 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
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
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
This is an example showing how to simplify an expression into scientific notation.
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
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
Learn how to solve a word problem by writing an equation to model the situation. In this video, we use the linear equation 210(t5) = 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
Students will practice two step equations, some of which require combining like terms and using the distributive property.
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
Given a graph, we will be able to find the equation it represents.
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
This tutorial will help you to explore slopes of lines and see how slope is represented on the xy axes.
Type: Tutorial
This tutorial reviews the concept of exponents and powers and includes how to evaluate powers with negative signs.
Type: Tutorial
This tutorial demonstrates how to use the power of a power property with both numerals and variables.
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 short video explains how to solve multistep equations with variables on both sides and why it is necessary to complete the same steps on both sides of the equation.
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
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 nonvertical 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
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 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 twostep 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
Video/Audio/Animations
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
This 5minute 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
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
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
"Slope" is a fundamental concept in mathematics. Slope of a linear function is often defined as " the rise over the run"....but why?
Type: Video/Audio/Animation
Linear equations can be used to solve many types of realword 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
Khan Academy tutorial video that demonstrates with realworld 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
Virtual Manipulatives
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 slopeintercept form or standard form. This activity allows students to explore linear equations, slopes, and yintercepts 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 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(yintercept) 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
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
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 is an online graphing utility that can be used to create box plots, bubble graphs, scatterplots, histograms, and stemandleaf 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
With a mouse, students will drag data points (with their error bars) and watch the bestfit 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 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
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