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Course Standards
General Course Information and Notes
General Notes
English Language Development ELD Standards Special Notes Section:
Teachers are required to provide listening, speaking, reading and writing instruction that allows English language learners (ELL) to communicate information, ideas and concepts for academic success in the content area of Mathematics. For the given level of English language proficiency and with visual, graphic, or interactive support, students will interact with grade level words, expressions, sentences and discourse to process or produce language necessary for academic success. The ELD standard should specify a relevant content area concept or topic of study chosen by curriculum developers and teachers which maximizes an ELL’s need for communication and social skills. To access an ELL supporting document which delineates performance definitions and descriptors, please click on the following link: https://cpalmsmediaprod.blob.core.windows.net/uploads/docs/standards/eld/ma.pdf.
General Information
Educator Certifications
Student Resources
Original Student Tutorials
Use the Pythagorean Theorem to find the legs of a right triangle in mathematical and real worlds contexts in this interactive tutorial.
This is part 3 in a 3-part series. Click below to explore the other tutorials in the series.
Type: Original Student Tutorial
Use the Pythagorean Theorem to find the hypotenuse of a right triangle in mathematical and real worlds contexts in this interactive tutorial.
This is part 2 in a 3-part series. Click below to explore the other tutorials in the series.
Type: Original Student Tutorial
Learn what the Pythagorean Theorem and its converse mean, and what Pythagorean Triples are in this interactive tutorial.
This is part 1 in a 3-part series. Click below to explore the other tutorials in the series.
Type: Original Student Tutorial
Learn how to simplify radicals in this interactive tutorial.
Type: Original Student Tutorial
Roll up your sleeves and learn how proportions can be used in everyday life in this interactive tutorial.
Type: Original Student Tutorial
Learn what non-perfect squares are and find the decimal approximation of their square roots in this interactive tutorial.
Type: Original Student Tutorial
Learn what perfect squares are and find their square roots in this interactive tutorial.
Type: Original Student Tutorial
Follow Hailey and Kenna as they estimate tips and sales tax at the mall, restaurants, and the hair salon in this interactive tutorial.
Type: Original Student Tutorial
Let's calculate markups and markdowns at the mall and follow Paige and Miriam working in this interactive tutorial.
Type: Original Student Tutorial
Calculate simple interest and estimate monthly payments alongside a loan officer named Jordan in this interactive tutorial.
Type: Original Student Tutorial
Explore sales tax, fees, and commission by following a customer service representative named Julian in this interactive tutorial.
Type: Original Student Tutorial
Learn to solve percent change problems involving percent increases and decreases in in this interactive tutorial.
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
Use models to solve balance problems on a space station in this interactive, math and science 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 combine like terms to create equivalent expressions in this cooking-themed, interactive tutorial.
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
Practice writing different aspects of an expository essay about scientists using drones to research glaciers in Peru. This interactive tutorial is part four of a four-part series. In this final tutorial, you will learn about the elements of a body paragraph. You will also create a body paragraph with supporting evidence. Finally, you will learn about the elements of a conclusion and practice creating a “gift.”
This tutorial is part four of a four-part series. Click below to open the other tutorials in this series.
- Drones and Glaciers: Eyes in the Sky (Part 1)
- Drones and Glaciers: Eyes in the Sky (Part 2)
- Expository Writing: Eyes in the Sky (Part 3)
- Expository Writing: Eyes in the Sky (Part 4)
Type: Original Student Tutorial
Learn how to write an introduction for an expository essay in this interactive tutorial. This tutorial is the third part of a four-part series. In previous tutorials in this series, students analyzed an informational text and video about scientists using drones to explore glaciers in Peru. Students also determined the central idea and important details of the text and wrote an effective summary. In part three, you'll learn how to write an introduction for an expository essay about the scientists' research.
This tutorial is part three of a four-part series. Click below to open the other tutorials in this series.
- Drones and Glaciers: Eyes in the Sky (Part 1)
- Drones and Glaciers: Eyes in the Sky (Part 2)
- Expository Writing: Eyes in the Sky (Part 3)
- Expository Writing: Eyes in the Sky (Part 4)
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
Compare and contrast mitosis and meiosis in this interactive tutorial. You'll also relate them to the processes of sexual and asexual reproduction and their consequences for genetic variation.
Type: Original Student Tutorial
Learn how to explain the meaning of additive inverse, identify the additive inverse of a given rational number, and justify your answer on a number line in this original tutorial.
Type: Original Student Tutorial
Learn to solve problems involving the circumference and area of circle-shaped pools in this interactive tutorial.
Type: Original Student Tutorial
Explore the relationship between mutations, the cell cycle, and uncontrolled cell growth which may result in cancer with this interactive tutorial.
Type: Original Student Tutorial
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
Educational Games
Play this interactive game and determine whether the similar shapes have gone through rotations, translations, or reflections.
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
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
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
Perspectives Video: Experts
A math teacher describes the relationship between area and circumference and gives examples in nature.
Type: 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/Enthusiasts
Understand 3D modeling from a new angle when you learn about surface geometry and 3D printing.
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
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
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
This problem solving task asks students to explain which measurements are needed to estimate the thickness of a soda can. Multiple solution processes are presented.
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
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
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
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
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
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 a two-step 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 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 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
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 .
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
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
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. Students are also asked to write an equation and graph each scenario.
Type: Problem-Solving Task
Students will answer questions about unit price of coffee, make a graph of the information, and explain the meaning of constant of proportionality/slope in the given context.
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 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
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
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 this resource, students experiment with the reflection of a triangle in a coordinate plane.
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 asks students to apply the effect of a single transformation on two-dimensional figures. 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 two-dimensional figures are 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. 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.
Type: Problem-Solving Task
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 the choice of representation. For example, 0.333¯ and 1/3 are two different ways of representing the same number.
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 can express a given repeating decimal as a fraction. Teachers looking for a task to fill in this background knowledge could consider the related task "Converting Decimal Representations of Rational Numbers to Fraction Representations".
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
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, 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 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.
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
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
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
This is an instructional task meant to generate a conversation around the meaning of negative integer exponents. It is good for students to learn the convention that negative time is simply any time before t=0.
Type: Problem-Solving Task
Students are asked to solve an inequality in order to answer a real-world question.
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
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
This video demonstrates the effect of a dilation on the coordinates of a triangle.
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 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
This video shows some examples that test your understanding of what happens when positive and negative numbers are multiplied and divided.
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
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 tutorial, you will practice classifying numbers as whole numbers, integers, rational numbers, and irrational numbers.
Type: Tutorial
This video discusses exponent properties involving products.
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 tutorial shows students the rule for negative exponents. Students will see, using variables, the pattern for negative exponents.
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
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 video shows how the area and circumference relate to each other and how changing the radius of a circle affects the area and circumference.
Type: Tutorial
In this video, students are shown the parts of a circle and how the radius, diameter, circumference and Pi relate to each other.
Type: Tutorial
This video shows how to find the circumference, the distance around a circle, given the area.
Type: Tutorial
In this video, watch as we find the area of a circle when given the diameter.
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 introductory video demonstrates the basic skill of how to write and solve a basic equation for a proportional relationship.
Type: Tutorial
In this example, we will work with three numbers in different formats: a percent, a decimal, and a mixed number.
Type: Tutorial
In this video, you will practice changing a fraction into decimal form.
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
This video shows how to recognize and understand graphs of proportional relationships to find the constant of proportionality.
Type: Tutorial
This introductory video teaches about combining like terms in linear 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
Find the volume of an object, given dimensions of a rectangular prism filled with water, and the incremental volume after the object is dropped into the water.
Type: Tutorial
This video involves packing a larger rectangular prism with smaller ones which is solved in two different ways.
Type: Tutorial
This video will show to find the volume of a triangular prism, and a cube by applying the formula for volume.
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
Here's an introductory video explaining the basic reasoning behind solving proportions and shows three different methods for solving proportions which you will use later on to solve more difficult problems.
Type: Tutorial
This introductory video shows some basic examples of writing two ratios and setting them equal to each other. This is just step 1 when solving word problems with proportions.
Type: Tutorial
This video demonstrates finding a unit rate from a rate containing fractions.
Type: Tutorial
Watch as we solve a rate problem finding speed in meters per second using distance (in meters) and time (in seconds).
Type: Tutorial
This video demonstrates adding and subtracting decimals in the context of an overdrawn checking account.
Type: Tutorial
The video will solve the inequality and graph the solution.
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
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
In this tutorial, you will use the distributive property to understand why the product of two negative numbers is positive.
Type: Tutorial
Practice substituting positive and negative values for variables.
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
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
This tutorial will help you to explore slopes of lines and see how slope is represented on the x-y axes.
Type: Tutorial
Learn how to find the full price when you know the discount price in this percent word problem.
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
Find out why subtracting a negative number is the same as adding the absolute value of that number.
Type: Tutorial
This video demonstrates adding and subtracting integers using a number line.
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
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 provides assistance with understanding direct and inverse variation.
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
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
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
A look behind the fundamental properties of the most basic arithmetic operation, addition
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 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
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.
Note: This video exceeds basic expectations for the mathematical concept(s) at this grade level. The video is intended for students who have demonstrated mastery within the scope of instruction who may be ready for a more rigorous extension of the mathematical concept(s). As with all materials, ensure to gauge the readiness of students or adapt according to student's needs prior to administration.
Type: Video/Audio/Animation
The video explains the process of creating linear equations to solve real-world problems.
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
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
"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
This Khan Academy video tutorial introduces averages and algebra problems involving averages.
Type: Video/Audio/Animation
Virtual Manipulatives
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
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 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 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 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
Congratulations
