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Course Standards
General Course Information and Notes
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Building on their work with linear, quadratic, and exponential functions, students extend their repertoire of functions to include polynomial, rational, and radical functions. Students work closely with the expressions that define the functions, and continue to expand and hone their abilities to model situations and to solve equations, including solving quadratic equations over the set of complex numbers and solving exponential equations using the properties of logarithms. The Standards for Mathematical Practice apply throughout each course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations. The critical areas for this course, organized into five units, are as follows:Unit 1- Polynomial, Rational, and Radical Relationships:This unit develops the structural similarities between the system of polynomials and the system of integers. Students draw on analogies between polynomial arithmetic and base-ten computation, focusing on properties of operations, particularly the distributive property. Students connect multiplication of polynomials with multiplication of multi-digit integers, and division of polynomials with long division of integers. Students identify zeros of polynomials, including complex zeros of quadratic polynomials, and make connections between zeros of polynomials and solutions of polynomial equations. The unit culminates with the fundamental theorem of algebra. A central theme of this unit is that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers.
Unit 2- Trigonometric Functions: Building on their previous work with functions, and on their work with trigonometric ratios and circles in Geometry, students now use the coordinate plane to extend trigonometry to model periodic phenomena.
Unit 3- Modeling with Functions: In this unit students synthesize and generalize what they have learned about a variety of function families. They extend their work with exponential functions to include solving exponential equations with logarithms. They explore the effects of transformations on graphs of diverse functions, including functions arising in an application, in order to abstract the general principle that transformations on a graph always have the same effect regardless of the type of the underlying function. They identify appropriate types of functions to model a situation, they adjust parameters to improve the model, and they compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit. The description of modeling as “the process of choosing and using mathematics and statistics to analyze empirical situations, to understand them better, and to make decisions” is at the heart of this unit. The narrative discussion and diagram of the modeling cycle should be considered when knowledge of functions, statistics, and geometry is applied in a modeling context.
Unit 4- Inferences and Conclusions from Data: In this unit, students see how the visual displays and summary statistics they learned in earlier grades relate to different types of data and to probability distributions. They identify different ways of collecting data— including sample surveys, experiments, and simulations—and the role that randomness and careful design play in the conclusions that can be drawn.
Unit 5- Applications of Probability: Building on probability concepts that began in the middle grades, students use the languages of set theory to expand their ability to compute and interpret theoretical and experimental probabilities for compound events, attending to mutually exclusive events, independent events, and conditional probability. Students should make use of geometric probability models wherever possible. They use probability to make informed decisions.
General Notes
Fluency RecommendationsA-APR.6 This standard sets an expectation that students will divide polynomials with remainder by inspection in simple cases. For example, one can view the rational expression
. A-SSE.2 The ability to see structure in expressions and to use this structure to rewrite expressions is a key skill in everything from advanced factoring (e.g., grouping) to summing series to the rewriting of rational expressions to examine the end behavior of the corresponding rational function.
F- IF.3 Fluency in translating between recursive definitions and closed forms is helpful when dealing with many problems involving sequences and series, with applications ranging from fitting functions to tables to problems in finance.
General Information
Educator Certifications
Student Resources
Original Student Tutorials
Learn how to use multistep factoring to factor quadratics in this interactive tutorial.
This is part 5 in a five-part series. Click below to open the other tutorials in this series.
- Part 1: The Diamond Game: Factoring Quadratics when a = 1
- Part 2: Factoring Polynomials Using Special Cases
- Part 3: Factoring Quadratics When the Coefficient a Does Not Equal 1: The Box Method
- Part 4: Factoring Polynomials when a Does Not Equal 1: Snowflake Method
- Part 5: Multistep Factoring: Quadratics (current tutorial)
Type: Original Student Tutorial
Learn to solve word problems represented by systems of linear equations, algebraically and graphically, in this interactive tutorial.
This part 7 in a 7-part series. Click below to explore the other tutorials in the series.
- Part 1: Solving Systems of Linear Equations: Using Graphs
- Part 2: Solving Systems of Linear Equations: Substitution
- Part 3: Solving Systems of Linear Equations: Basic Elimination
- Part 4: Solving Systems of Linear Equations: Advanced Elimination
- Part 5: Solving Systems of Linear Equations: Connecting Algebraic Methods to Graphing
- Part 6: Solving Systems of Linear Equations: Writing Systems from Context
Type: Original Student Tutorial
Learn to factor quadratic trinomials when the coefficient a does not equal 1 by using the Snowflake Method in this interactive tutorial.
This is part 4 in a five-part series. Click below to open the other tutorials in this series.
- Part 1: The Diamond Game: Factoring Quadratics when a = 1
- Part 2: Factoring Polynomials Using Special Cases
- Part 3: Factoring Quadratics When the Coefficient a Does Not Equal 1: The Box Method
- Part 4: Factoring Polynomials when a Does Not Equal 1: Snowflake Method (Current Tutorial)
- Part 5: Multistep Factoring: Quadratics
Type: Original Student Tutorial
Learn how to create systems of linear equations to represent contextual situations in this interactive tutorial.
This part 6 in a 7-part series. Click below to explore the other tutorials in the series.
- Part 1: Solving Systems of Linear Equations: Using Graphs
- Part 2: Solving Systems of Linear Equations: Substitution
- Part 3: Solving Systems of Linear Equations: Basic Elimination
- Part 4: Solving Systems of Linear Equations: Advanced Elimination
- Part 5: Solving Systems of Linear Equations: Connecting Algebraic Methods to Graphing
- Part 7: Solving Systems of Linear Equations: Word Problems (Coming soon)
Type: Original Student Tutorial
Learn to solve systems of linear equations by connecting algebraic and graphing methods in this interactive tutorial.
This part 5 in a 7-part series. Click below to explore the other tutorials in the series.
- Part 1: Solving Systems of Linear Equations: Using Graphs
- Part 2: Solving Systems of Linear Equations: Substitution
- Part 3: Solving Systems of Linear Equations: Basic Elimination
- Part 4: Solving Systems of Linear Equations: Advanced Elimination
- Part 6: Solving Systems of Linear Equations: Writing Systems from Context (Coming soon)
- Part 7: Solving Systems of Linear Equations: Word Problems (Coming soon)
Type: Original Student Tutorial
Learn how to factor quadratic polynomials when the leading coefficient (a) is not 1 by using the box method in this interactive tutorial.
This is part 3 in a five-part series. Click below to open the other tutorials in this series.
- Part 1: The Diamond Game: Factoring Quadratics when a = 1
- Part 2: Factoring Polynomials Using Special Cases
- Part 3: Factoring Quadratics When the Coefficient a Does Not Equal 1: The Box Method (Current Tutorial)
- Part 4: Factoring Polynomials when a Does Not Equal 1: Snowflake Method
- Part 5: Multistep Factoring: Quadratics
Type: Original Student Tutorial
Learn how to factor quadratics when the coefficient a = 1 using the diamond method in this game show-themed, interactive tutorial.
This is part 1 in a five-part series. Click below to open the other tutorials in this series.
- Part 1: The Diamond Game: Factoring Quadratics when a = 1 (Current Tutorial)
- Part 2: Factoring Polynomials Using Special Cases
- Part 3: Factoring Quadratics When the Coefficient a Does Not Equal 1: The Box Method
- Part 4: Factoring Polynomials when a Does Not Equal 1: Snowflake Method
- Part 5: Multistep Factoring: Quadratics
Type: Original Student Tutorial
Learn to solve systems of linear equations using advanced elimination in this interactive tutorial.
This part 4 in a 7-part series. Click below to explore the other tutorials in the series.
- Part 1: Solving Systems of Linear Equations Part 1: Using Graphs
- Part 2: Solving Systems of Linear Equations Part 2: Substitution
- Part 3: Solving Systems of Linear Equations Part 3: Basic Elimination
- Part 5: Solving Systems of Linear Equations Part 5: Connecting Algebraic Methods to Graphing (Coming soon)
- Part 6: Solving Systems of Linear Equations Part 6: Writing Systems from Context (Coming soon)
- Part 7: Solving Systems of Linear Equations Part 7: Word Problems (Coming soon)
Type: Original Student Tutorial
Learn to solve systems of linear equations using basic elimination in this interactive tutorial.
This part 3 in a 7-part series. Click below to explore the other tutorials in the series.
Part 1: Solving Systems of Linear Equations Part 1: Using Graphs
Part 2: Solving Systems of Linear Equations Part 2: Substitution
Part 4: Solving Systems of Linear Equations Part 4: Advanced Elimination (Coming soon)
Part 5: Solving Systems of Linear Equations Part 5: Connecting Algebraic Methods to Graphing (Coming soon)
Part 6: Solving Systems of Linear Equations Part 6: Writing Systems from Context (Coming soon)
Part 7: Solving Systems of Linear Equations Part 7: Word Problems (Coming soon)
Type: Original Student Tutorial
Learn the process of completing the square of a quadratic function to find the maximum or minimum to discover how high a dolphin jumped in this interactive tutorial.
This is part 2 of a 2 part series. Click HERE to open part 1.
Type: Original Student Tutorial
Learn the process of completing the square of a quadratic function to find the maximum or minimum to discover how high a dolphin jumped in this interactive tutorial.
This is part 1 of a 2 part series. Click HERE to open Part 2.
Type: Original Student Tutorial
Learn to identify and interpret parts of linear expressions in terms of mathematical or real-world contexts in this original tutorial.
Type: Original Student Tutorial
Learn to solve systems of linear equations using substitution in this interactive tutorial.
This part 2 in a 7-part series. Click below to explore the other tutorials in the series.
Part 1: Solving Systems of Linear Equations Part 1: Using Graphs
Part 3: Solving Systems of Linear Equations Part 3: Basic Elimination (Coming soon)
Part 4: Solving Systems of Linear Equations Part 4: Advanced Elimination (Coming soon)
Part 5: Solving Systems of Linear Equations Part 5: Connecting Algebraic Methods to Graphing (Coming soon)
Part 6: Solving Systems of Linear Equations Part 6: Writing Systems from Context (Coming soon)
Part 7: Solving Systems of Linear Equations Part 7: Word Problems (Coming soon)
Type: Original Student Tutorial
Learn about exponential decay as you calculate the value of used cars by examining equations, graphs, and tables in this interactive tutorial.
Type: Original Student Tutorial
Learn how to solve systems of linear equations graphically in this interactive tutorial.
Type: Original Student Tutorial
Learn how to interpret key features of linear functions and translate between representations of linear functions through exploring jobs for teenagers in this interactive tutorial.
Type: Original Student Tutorial
Learn about exponential growth in the context of interest earned as money is put in a savings account by examining equations, graphs, and tables in this interactive tutorial.
Type: Original Student Tutorial
Learn about exponential functions and how they are different from linear functions by examining real world situations, their graphs and their tables in this interactive tutorial.
Type: Original Student Tutorial
Learn how to solve rational functions by getting common denominators in this interactive tutorial.
Type: Original Student Tutorial
Visualize the effect of using a value of k in both kf(x) or f(kx) when k is greater than zero in this interactive tutorial.
Type: Original Student Tutorial
Learn how to solve rational linear and quadratic equations using cross multiplication in this interactive tutorial.
Type: Original Student Tutorial
Learn how to solve and graph compound inequalities and determine if solutions are viable in part 2 of this interactive tutorial series.
Click HERE to open Part 1.
Type: Original Student Tutorial
Learn how to write equations in two variables in this interactive tutorial.
Type: Original Student Tutorial
Learn how to solve and graph one variable inequalities, including compound inequalities, in part 1 of this interactive tutorial series.
Click HERE to open Part 2.
Type: Original Student Tutorial
Learn how reflections of a function are created and tied to the value of k in the mapping of f(x) to -1f(x) in this interactive tutorial.
Type: Original Student Tutorial
Explore translations of functions on a graph that are caused by k in this interactive tutorial. GeoGebra and interactive practice items are used to investigate linear, quadratic, and exponential functions and their graphs, and the effect of a translation on a table of values.
Type: Original Student Tutorial
This is Part Two of a two-part series. Learn to identify faulty reasoning in this interactive tutorial series. You'll learn what some experts say about year-round schools, what research has been conducted about their effectiveness, and how arguments can be made for and against year-round education. Then, you'll read a speech in favor of year-round schools and identify faulty reasoning within the argument, specifically the use of hasty generalizations.
Make sure to complete Part One before Part Two! Click HERE to launch Part One.
Type: Original Student Tutorial
Learn to identify faulty reasoning in this two-part interactive English Language Arts tutorial. You'll learn what some experts say about year-round schools, what research has been conducted about their effectiveness, and how arguments can be made for and against year-round education. Then, you'll read a speech in favor of year-round schools and identify faulty reasoning within the argument, specifically the use of hasty generalizations.
Make sure to complete both parts of this series! Click HERE to open Part Two.
Type: Original Student Tutorial
Examine President John F. Kennedy's inaugural address in this interactive tutorial. You will examine Kennedy's argument, main claim, smaller claims, reasons, and evidence.
In Part Four, you'll use what you've learned throughout this series to evaluate Kennedy's overall argument.
Make sure to complete the previous parts of this series before beginning Part 4.
Type: Original Student Tutorial
Examine President John F. Kennedy's inaugural address in this interactive tutorial. You will examine Kennedy's argument, main claim, smaller claims, reasons, and evidence. By the end of this four-part series, you should be able to evaluate his overall argument.
In Part Three, you will read more of Kennedy's speech and identify a smaller claim in this section of his speech. You will also evaluate this smaller claim's relevancy to the main claim and evaluate Kennedy's reasons and evidence.
Make sure to complete all four parts of this series!
Type: Original Student Tutorial
Learn to find the zeros of a quadratic function and interpret their meaning in real-world contexts with this interactive tutorial.
Type: Original Student Tutorial
Learn how to add and subtract polynomials in this online tutorial. You will learn how to combine like terms and then use the distribute property to subtract polynomials.
This is part 2 of a two-part lesson. Click below to open part 1.
Type: Original Student Tutorial
Learn how to identify monomials and polynomials and determine their degree in this interactive tutorial.
This is part 1 in a two-part series. Click here to open Part 2.
Type: Original Student Tutorial
This is Part Two of a two-part tutorial series. In this interactive tutorial, you'll practice identifying a speaker's purpose using a speech by aviation pioneer Amelia Earhart. You will examine her use of rhetorical appeals, including ethos, logos, pathos, and kairos. Finally, you'll evaluate the effectiveness of Earhart's use of rhetorical appeals.
Be sure to complete Part One first. Click here to launch PART ONE.
Type: Original Student Tutorial
This is Part One of a two-part tutorial series. In this interactive tutorial, you'll practice identifying a speaker's purpose using a speech by aviation pioneer Amelia Earhart. You will examine her use of rhetorical appeals, including ethos, logos, pathos, and kairos. Finally, you'll evaluate the effectiveness of Earhart's use of rhetorical appeals.
Click here to launch PART TWO.
Type: Original Student Tutorial
Use long division to rewrite a rational expression of the form a(x) divided by b(x) in the form q(x) plus the quantity r(x) divided by b(x), where a(x), b(x), q(x), and r(x) are polynomials with this interactive tutorial.
Type: Original Student Tutorial
Learn to complete the square of a quadratic expression and identify the maximum or minimum value of the quadratic function it defines. In this interactive tutorial, you'll also interpret the meaning of the maximum and minimum of a quadratic function in a real world context.
Type: Original Student Tutorial
Learn to rewrite products involving radicals and rational exponents using properties of exponents in this interactive tutorial.
Type: Original Student Tutorial
Learn how to calculate and interpret an average rate of change over a specific interval on a graph in this interactive tutorial.
Type: Original Student Tutorial
Learn how to explain the steps used to solve multi-step linear equations and provide reasons to support those steps with this interactive tutorial.
Type: Original Student Tutorial
Follow as we learn why the x-coordinate of the point of intersection of two functions is the solution of the equation f(x) = g(x) in this interactive tutorial.
Type: Original Student Tutorial
Learn about the radian measures of an angle, finding an angle measure in radians given the arc length and length of the radius, and converting between degree measures and radian measures in this interactive tutorial.
Type: Original Student Tutorial
Follow as we discover key features of a quadratic equation written in vertex form in this interactive tutorial.
Type: Original Student Tutorial
Write linear inequalities for different money situations in this interactive tutorial.
Type: Original Student Tutorial
Learn how to factor quadratic polynomials that follow special cases, difference of squares and perfect square trinomials, in this interactive tutorial.
This is part 2 in a five-part series. Click below to open the other tutorials in this series.
- Part 1: The Diamond Game: Factoring Quadratics when a = 1
- Part 2: Factoring Polynomials Using Special Cases (Current Tutorial)
- Part 3: Factoring Quadratics When the Coefficient a Does Not Equal 1: The Box Method
- Part 4: Factoring Polynomials when a Does Not Equal 1: Snowflake Method
- Part 5: Multistep Factoring: Quadratics
Type: Original Student Tutorial
Educational Games
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 Excel spreadsheet allows the educator to input data into a two way frequency table and have the resulting relative frequency charts calculated automatically on the second sheet. This resource will assist the educator in checking student calculations on student-generated data quickly and easily.
Steps to add data: All data is input on the first spreadsheet; all tables are calculated on the second spreadsheet
- Modify column and row headings to match your data.
- Input joint frequency data.
- Click the second tab at the bottom of the window to see the automatic calculations.
Type: Educational Software / Tool
Perspectives Video: Experts
Jump to it and learn more about how quadratic equations are used in robot navigation problem solving!
Type: Perspectives Video: Expert
The tide is high! How can we statistically prove there is a relationship between the tides on the Gulf Coast and in a fresh water spring 20 miles from each other?
Download the CPALMS Perspectives video student note taking guide.
Type: Perspectives Video: Expert
Statistical analysis played an essential role in using microgravity sensors to determine location of caves in Wakulla County.
Download the CPALMS Perspectives video student note taking guide.
Type: Perspectives Video: Expert
It's important to stay inside the lines of your project constraints to finish in time and under budget. This NASA systems engineer explains how constraints can actually promote creativity and help him solve problems!
Type: Perspectives Video: Expert
The director of the National High Magnetic Field Laboratory describes electromagnetic waves.
Download the CPALMS Perspectives video student note taking guide.
Type: Perspectives Video: Expert
Perspectives Video: Professional/Enthusiasts
Listen in as a computing enthusiast describes how hexadecimal notation is used to express big numbers in just a little space.
Download the CPALMS Perspectives video student note taking guide.
Type: Perspectives Video: Professional/Enthusiast
No need to sugar coat it: making candy involves math and muscles. Learn how light refraction and exponential growth help make candy colors just right!
Type: Perspectives Video: Professional/Enthusiast
This buzzworthy video features statistics, sampling, and how scientists make inferences about populations.
Type: Perspectives Video: Professional/Enthusiast
Get in gear with robotics as this engineer explains how quadratic equations are used in programming robotic navigation.
Type: Perspectives Video: Professional/Enthusiast
Problem-Solving Tasks
Students explore the structure of the operation s/(vn). This question provides students with an opportunity to see expressions as constructed out of a sequence of operations: first taking the square root of n, then dividing the result of that operation into s.
Type: Problem-Solving Task
This problem solving task challenges students to determine if two weather events are independent, and use that conclusion to find the probability of having similar weather events under certain conditions.
Type: Problem-Solving Task
The task is intended to address sample space, independence, probability distributions and permutations/combinations.
Type: Problem-Solving Task
Students answer questions about the probabilities of independent and dependent events.
Type: Problem-Solving Task
This problem solving task lets students explore the concept of independence of events.
Type: Problem-Solving Task
This task assesses a student's ability to use the addition rule to compute a probability and to interpret a probability in context.
Type: Problem-Solving Task
The purpose of this task is to assess a student's ability to explain the meaning of independence in a simple context.
Type: Problem-Solving Task
This problem solving task asks students to make deductions about the kind of music students enjoy by examining data in a two-way table.
Type: Problem-Solving Task
Students are asked to choose the best sampling method for choosing the new School Advisory Panel.
Type: Problem-Solving Task
The purpose of this task is to assess (1) ability to distinguish between an observational study and an experiment and (2) understanding of the role of random assignment to experimental groups in an experiment.
Type: Problem-Solving Task
This problem solving task challenges students to answer probability questions about SAT scores, using distribution and mean to solve the problem.
Type: Problem-Solving Task
The purpose of this task is to have students complete normal distribution calculations and to use properties of normal distributions to draw conclusions.
Type: Problem-Solving Task
This task requires students to use the normal distribution as a model for a data distribution. Students must use given means and standard deviations to approximate population percentages.
Type: Problem-Solving Task
This task lets students explore the concepts of probability as a fraction of outcomes using two-way tables.
Type: Problem-Solving Task
This task asks students to calculate probabilities using information presented in a two-way frequency table.
Type: Problem-Solving Task
This problem solving task asks students to determine probabilities and draw conclusions about the survival rates on the Titanic using a table of data.
Type: Problem-Solving Task
In this example, students are asked to write a function describing the population growth of algae. It is implied that this is exponential growth.
Type: Problem-Solving Task
This problem solving task challenges students to use trigonometric functions to model the number of rabbits and foxes as a function of time.
Type: Problem-Solving Task
This problem solving task challenges students to trigonometric functions to model the populations of rabbits and foxes over time, and then graph the functions.
Type: Problem-Solving Task
In this task, students use trigonometric functions to model the movement of a point around a wheel and, through space. Students also interpret features of graphs in terms of the given real-world context.
Type: Problem-Solving Task
This problem solving task asks students to predict and model US population based on a chart of US population data from 1982 to 1988.
Type: Problem-Solving Task
This simple conceptual problem does not require algebraic manipulation, but requires students to articulate the reasoning behind each statement. Students are asked to verify a given linear equation from data in a table and interpret its key components in context.
Type: Problem-Solving Task
The purpose of this task is to give students experience modeling a real-world example of exponential growth, in a context that provides a vivid illustration of the power of exponential growth, for example the cost of inaction for a year. There is an opportunity for further discussion based on part (c), since the ratio of costs from one year to the next is the same in each part.
Type: Problem-Solving Task
The coffee cooling experiment is a popular example of an exponential model with immediate appeal. The model is realistic and provides a good context for students to practice work with exponential equations.
Type: Problem-Solving Task
Although this task is fairly straightforward, it is worth noticing that it does not explicitly tell students to look for intersection points when they graph the circle and the line. Thus, in addition to assessing whether they can solve the system of equations, it is assessing a simple but important piece of conceptual understanding, namely the correspondence between intersection points of the two graphs and solutions of the system.
Type: Problem-Solving Task
In order to engage this task meaningfully, students must be aware of the convention that va for a positive number a refers to the positive square root of a. The purpose of the task is to show students a situation where squaring both sides of an equation can result in an equation with more solutions than the original one.
Type: Problem-Solving Task
This mathematical modeling task also illustrates making sense of a problem. Students are only told that there are two ingredients in the pasta and they have a picture of the box. It might even be better to just show the picture of the box, or to bring in the box and ask the students to pose the question themselves. The brand of pasta is quite commonly available at supermarkets or health food stores such as Whole Foods and even at Amazon.com. The box has the nutritional label and a reference to the website where the students can find other information about the ingredients
Type: Problem-Solving Task
This mathematical modeling task also illustrates making sense of a problem. Students are given all the relevant information on the nutritional labels, but they have to figure out how to use this information. They have to come up with the idea that they can set up two equations in two unknowns to solve the problem.
Type: Problem-Solving Task
This task addresses solving systems of linear equations, and provides a simple example of a system with three equations and three unknown. Two (of many) methods for solving the system are presented. The first takes the given information to make three equations in three unknowns which can then be solved via algebraic manipulation to find the three numbers. The second solution is more clever, creating a single equation in three unknowns from the given information. This equation is then combined with the given information about the sums of pairs of numbers to deduce what the third number is. In reality, this solution is not simpler than the first: rather it sets up a slightly different set of equations which can be readily solved (the key being to take the sum of the three equations in the first solution). It provides a good opportunity for the instructor to show different methods for solving the same system of linear equations.
Type: Problem-Solving Task
The purpose of this task is to continue a crucial strand of algebraic reasoning begun at the middle school level (e.g, 6.EE.5). By asking students to reason about solutions without explicitly solving them, we get at the heart of understanding what an equation is and what it means for a number to be a solution to an equation. The equations are intentionally very simple; the point of the task is not to test technique in solving equations, but to encourage students to reason about them.
Type: Problem-Solving Task
This problem introduces the method used by scientists to date certain organic material. It is based not on the amount of the Carbon 14 isotope remaining in the sample but rather on the ratio of Carbon 14 to Carbon 12. This ratio decreases, hypothetically, at a constant exponential rate as soon as the organic material has ceased to absorb Carbon 14, that is, as soon as it dies.
Type: Problem-Solving Task
In the task "Carbon 14 Dating" the amount of Carbon 14 in a preserved plant is studied as time passes after the plant has died. In practice, however, scientists wish to determine when the plant died, and as this task shows, that is not possible with a simple measurement of the amount of Carbon 14 remaining in the preserved plant.
Type: Problem-Solving Task
The task requires the student to use logarithms to solve an exponential equation in the realistic context of carbon dating, important in archaeology and geology, among other places. Students should be guided to recognize the use of the natural logarithm when the exponential function has the given base of e, as in this problem. Note that the purpose of this task is algebraic in nature -- closely related tasks exist which approach similar problems from numerical or graphical stances.
Type: Problem-Solving Task
This task provides a real world context for interpreting and solving exponential equations. There are two solutions provided for part (a). The first solution demonstrates how to deduce the conclusion by thinking in terms of the functions and their rates of change. The second approach illustrates a rigorous algebraic demonstration that the two populations can never be equal.
Type: Problem-Solving Task
This task provides an interesting context to ask students to estimate values in an exponential function using a graph.
Type: Problem-Solving Task
In this task students use verbal descriptions to construct and compare linear and exponential functions and to find where the two functions intersect (F-LE.2, F-LE.3, A-REI.11).
Type: Problem-Solving Task
This task provides an exploration of a quadratic equation by descriptive, numerical, graphical, and algebraic techniques. Based on its real-world applicability, teachers could use the task as a way to introduce and motivate algebraic techniques like completing the square, en route to a derivation of the quadratic formula.
Type: Problem-Solving Task
This task asks students to consider the linear and quadratic functions shown on a graph, and use quadratic functions to find the coordinates.
Type: Problem-Solving Task
The given solutions for this task involve the creation and solving of a system of two equations and two unknowns, with the caveat that the context of the problem implies that we are interested only in non-negative integer solutions. Indeed, in the first solution, we must also restrict our attention to the case that one of the variables is further even. This aspect of the task is illustrative of the mathematical practice of modeling with mathematics, and crucial as the system has an integer solution for both situations, that is, whether we include the dollar on the floor in the cash box or not.
Type: Problem-Solving Task
This task is a somewhat more complicated version of "Accurately weighing pennies I'' as a third equation is needed in order to solve part (a) explicitly. Instead, students have to combine the algebraic techniques with some additional problem-solving (numerical reasoning, informed guess-and-check, etc.) Part (b) is new to this task, as with only two types of pennies the weight of the collection determines how many pennies of each type are in the collection. This is no longer the case with three different weights but in this particular case, a collection of 50 is too small to show any ambiguity. This is part of the reason for part (c) of the question where the weight alone no longer determines which type of pennies are in the roll. This shows how important levels of accuracy in measurement are as the answer to part (b) could be different if we were to measure on a scale which is only accurate to the nearest tenth of a gram instead of to the nearest hundredth of a gram.
Type: Problem-Solving Task
Solving this problem with algebra requires factoring a particular cubic equation (the difference of two cubes) as well as a quadratic equation. An alternative solution using prime numbers and arithmetic is presented.
Type: Problem-Solving Task
The purpose of this instructional task is to give students an opportunity to construct and find the value of a geometric series (A-SSE.4) in a financial literacy context. The task assumes that students have already developed the formula for a geometric series themselves; having them recognize the need for this formula (and look up if necessary) allows them to engage in MP 5, Use appropriate tools strategically. The task also provides students with an opportunity to look for and express regularity in repeated reasoning (MP 8), as the solution shows. This task also asks students to interpret the variables in the future value formula in the context of the problem (A-SSE.1).
Type: Problem-Solving Task
This classroom task is meant to elicit a variety of different methods of solving a quadratic equation (A-REI.4). Some are straightforward (for example, expanding the square on the right and rearranging the equation so that we can use the quadratic formula); some are simple but clever (reasoning from the fact that x and (2x - 9) have the same square); some use tools (using a graphing calculator to graph the functions f(x) = x^2 and g(x) = (2x-90)^2 and looking for values of x at which the two functions intersect). Some solution methods will work on an arbitrary quadratic equation, while others (such as the last three) may have difficulty or fail if the quadratic equation is not given in a particular form, or if the solutions are not rational numbers.
Type: Problem-Solving Task
This problem involves solving a system of algebraic equations from a context: depending how the problem is interpreted, there may be one equation or two. The main work in parts (a) and (b) is in setting up the equation(s) appropriately. Question (c) is more subtle and it requires thinking carefully about the accuracy available in a particular measurement (weight). The first two parts of this task could be used for instructional or assessment purposes while the third part should strictly be implemented for instructional purposes.
Type: Problem-Solving Task
The purpose of this task is to provide an opportunity for students to reason about equivalence of equations. The instruction to give reasons that do not depend on solving the equation is intended to focus attention on the transformation of equations as a deductive step.
Type: Problem-Solving Task
This problem-solving task asks students to describe exponential growth through a real-world problem involving the illegal introduction of fish into a lake.
Type: Problem-Solving Task
This problem-solving task challenges students to find all quadratic functions described by given equation and coordinates, and describe how the graphs of those functions are related to one another.
Type: Problem-Solving Task
This is a simple task touching on two key points of functions. First, there is the idea that not all functions have real numbers as domain and range values. Second, the task addresses the issue of when a function admits an inverse, and the process of "restricting the domain" in order to achieve an invertible function.
Type: Problem-Solving Task
The task addresses knowledge related to interpreting forms of functions derived by factoring or completing the square. It requires students to pay special attention to the information provided by the way the equation is represented as well as the sign of the leading coefficient, which is not written out explicitly, and then to connect this information to the important features of the graph.
Type: Problem-Solving Task
This task is meant to be a straight-forward assessment task of graph reading and interpreting skills. This task helps reinforce the idea that when a variable represents time, t = 0 is chosen as an arbitrary point in time and positive times are interpreted as times that happen after that.
Type: Problem-Solving Task
This task could be used for assessment or for practice. It allows students to compare characteristics of two quadratic functions that are each represented differently, one as the graph of a quadratic function and one written out algebraically. Specifically, students are asked to determine which function has the greatest maximum and the greatest non-negative root.
Type: Problem-Solving Task
This task asks students to find the average, write an equation, find the domain, and create a graph of the cost of producing DVDs.
Type: Problem-Solving Task
The problem presents a context where a quadratic function arises. Careful analysis, including graphing of the function, is closely related to the context. The student will gain valuable experience applying the quadratic formula and the exercise also gives a possible implementation of completing the square.
Type: Problem-Solving Task
The purpose of this task is to construct and use inverse functions to model a real-life context. Students choose a linear function to model the given data, and then use the inverse function to interpolate a data point.
Type: Problem-Solving Task
Temperature conversions provide a rich source of linear functions which are encountered not only in science but also in our every day lives when we travel abroad. The first part of this task provides an opportunity to construct a linear function given two input-output pairs. The second part investigates the inverse of a linear function while the third part requires reasoning about quantities and/or solving a linear equation.
Type: Problem-Solving Task
In this task students investigate and ultimately prove the validity of the method of generating Pythagorean Triples that involves the polynomial identity (x2+y2)2=(x2-y2)2+(2xy)2.
Type: Problem-Solving Task
This task asks students to consider functions in regard to temperatures in a high school gym.
Type: Problem-Solving Task
In this task students are given graphs of quantities related to weather. The purpose of the task is to show that graphs are more than a collection of coordinate points; they can tell a story about the variables that are involved, and together they can paint a very complete picture of a situation, in this case the weather. Features in one graph, like maximum and minimum points, correspond to features in another graph. For example, on a rainy day, the solar radiation is very low, and the cumulative rainfall graph is increasing with a large slope.
Type: Problem-Solving Task
This deceptively simple task asks students to find the domain and range of a function from a given context. The function is linear and if simply looked at from a formulaic point of view, students might find the formula for the line and say that the domain and range are all real numbers.
Type: Problem-Solving Task
This problem introduces a logistic growth model in the concrete settings of estimating the population of the U.S. The model gives a surprisingly accurate estimate and this should be contrasted with linear and exponential models.
Type: Problem-Solving Task
This task is for instructional purposes only and students should already be familiar with some specific examples of logistic growth functions. The goal of this task is to have students appreciate how different constants influence the shape of a graph.
Type: Problem-Solving Task
This task can be used as a quick assessment to see if students can make sense of a graph in the context of a real world situation. Students also have to pay attention to the scale on the vertical axis to find the correct match. The first and third graphs look very similar at first glance, but the function values are very different since the scales on the vertical axes are very different. The task could also be used to generate a group discussion on interpreting functions given by graphs.
Type: Problem-Solving Task
In this task, students will use inverse operations to solve the equations for the unknown variable or for the designated variable if there is more than one.
Type: Problem-Solving Task
The purpose of this task is to give students practice writing a constraint equation for a given context. Instruction accompanying this task should introduce the notion of a constraint equation as an equation governing the possible values of the variables in question (i.e., "constraining" said values). In particular, it is worth differentiating the role of constraint equations from more functional equations, e.g., formulas to convert from degrees Celsius to degree Fahrenheit. The task has students interpret the context and choose variables to represent the quantities, which are governed by the constraint equation and the fact that they are non-negative (allowing us to restrict the graphs to points in the first quadrant only).
The four parts are independent and can be used as separate tasks.
Type: Problem-Solving Task
This task presents a simple but mathematically interesting game whose solution is a challenging exercise in creating and reasoning with algebraic inequalities. The core of the task involves converting a verbal statement into a mathematical inequality in a context in which the inequality is not obviously presented, and then repeatedly using the inequality to deduce information about the structure of the game.
Type: Problem-Solving Task
In this task, students are asked to show or verify four theorems related to roots, zeroes, and factors of polynomial functions. The Fundamental theorem of Arithmetic is also mentioned. This task builds on "Zeroes and factorization of a quadratic function'' parts I and II.
Type: Problem-Solving Task
Students are given a word problem that can be solved by using a pair of linear equations. This task does not actually require that the student solve the system but that they recognize the pairs of linear equations in two variables that would be used to solve the system. This is an important step in the process of solving systems.
Type: Problem-Solving Task
This task examines the ways in which the plane can be covered by regular polygons in a very strict arrangement called a regular tessellation. These tessellations are studied here using algebra, which enters the picture via the formula for the measure of the interior angles of a regular polygon (which should therefore be introduced or reviewed before beginning the task). The goal of the task is to use algebra in order to understand which tessellations of the plane with regular polygons are possible.
Type: Problem-Solving Task
This task provides an approximation, and definition, of e, in the context of more and more frequent compounding of interest in a bank account. The approach is computational.
Type: Problem-Solving Task
For a polynomial function f, if f(0)=0 then the polynomial f(x) is divisible by x. This fact is shown and then generalized in "Zeroes of a quadratic polynomial I, II" and "Zeroes of a general polynomial.'' Here, divisibility tells us that the quotient f(x)/x will still be a nice function -- indeed, another polynomial, save for the missing point at x=0. The goal of this task is to show via a concrete example that this nice property of polynomials is not shared by all functions. The non-polynomial function F given by F(x)=|x| is a familiar function for which property does not hold: even though F(0)=0, the quotient F(x)/x behaves badly near x=0. Indeed, its graph is broken into two parts which do not connect at x=0.
Type: Problem-Solving Task
This task aims for students to understand the quadratic formula in a geometric way in terms of the graph of a quadratic function.
Type: Problem-Solving Task
This task is intended for instruction and to motivate "Building a general quadratic function." This task assumes that the students are familiar with the process of completing the square.
Type: Problem-Solving Task
In this example, students use properties of rational exponents and other algebraic concepts to compare and verify the relative size of two real numbers that involve decimal exponents.
Type: Problem-Solving Task
In this example, students are given the graph of two functions and are asked to sketch the graph of the function that is their sum. The intent is that students develop a conceptual understanding of function addition.
Type: Problem-Solving Task
Students consider a diagram of five nested equilateral triangles diminishing in size according to a geometric series. The purpose of this task is to emphasize the adjective "geometric" in the "geometric" series, namely, that the algebraic notion of a common ratio between terms corresponds to the geometric notion of a repeated similarity transformation. Specifically, since the black triangles are all similar with the same scale factor, the total area of the black triangles is a geometric series. This task could be used either to introduce the geometric series as a worthy object of study, or as a geometric application of its use.
Type: Problem-Solving Task
In this task, students consider a real-world problem involving the decay of a drug in a patient's body. This task presents a real world application of finite geometric series. The context can lead into several interesting follow-up questions and projects. Many drugs only become effective after the amount in the body builds up to a certain level. This can be modeled very well with geometric series.
Type: Problem-Solving Task
This task leads to the generation of finite geometric series with a common ratio less than one as a means to explore properties of the Cantor Set. The Cantor Set is a fascinating set with many intriguing properties. It contains uncountably many points, which means that there are "as many" points in it as on the real line, yet the set contains no intervals of real numbers and it has length zero. All that is necessary to show that it has length zero is to look at what happens to a geometric series as we add more and more terms.
Type: Problem-Solving Task
There are many different ways to write exponential expressions that describe the same quantity, in this task the amount of a radioactive substance after t years. Depending on what aspect of the context we need to investigate, one expression of the quantity may be more useful than another. This task contrasts the usefulness of four equivalent expressions. Students first have to confirm that the given expressions for the radioactive substance are equivalent. Then they have to explain the significance of each expression in the context of the situation.
Type: Problem-Solving Task
This is a challenging task, suitable for extended work, and reaching into a deep understanding of units. Students are given a scenario and asked to determine the number of people required to complete the amount of work in the time described. The task requires students to exhibit , Make sense of problems and persevere in solving them. An algebraic solution is possible but complicated; a numerical solution is both simpler and more sophisticated, requiring skilled use of units and quantitative reasoning. Thus the task aligns with either MAFS.912.A-CED.1.1 or MAFS.912.N-Q.1.1, depending on the approach.
Type: Problem-Solving Task
Students manipulate a given equation to find specified information.
Type: Problem-Solving Task
Students solve problems tracking the balance of a checking account used only to pay rent. This simple conceptual task focuses on what it means for a number to be a solution to an equation, rather than on the process of solving equations.
Type: Problem-Solving Task
Students extrapolate the list price of a car given a total amount paid in states with different tax rates. The emphasis in this task is not on complex solution procedures. Rather, the progression of equations, from two that involve different values of the sales tax, to one that involves the sales tax as a parameter, is designed to foster the habit of looking for regularity in solution procedures, so that students don't approach every equation as a new problem but learn to notice familiar types.
Type: Problem-Solving Task
This task addresses an important issue about inverse functions. In this case the function f is the inverse of the function g but g is not the inverse of f unless the domain of f is restricted.
Type: Problem-Solving Task
In this resource, students refer to given information which defines 5 variables in the context of real world government expenses. They are then asked to write equations based upon specific known values for some of the variables. The emphasis is on setting up, rather than solving, the equations.
Type: Problem-Solving Task
This task continues "Zeroes and factorization of a quadratic polynomial I.'' The argument here generalizes, as shown in "Zeroes and factorization of a general polynomial'' to show that a polynomial of degree d can have at most d roots. In the quadratic case, an alternative argument for why there can be at most two roots can be given using the quadratic formula and this is done in the second solution below.
This task is intended for instructional purposes to help students see more clearly the link between factorization of polynomials and zeroes of polynomial functions. Students who are familiar with the quadratic formula should be encouraged to think about the first solution which extends to polynomials of higher degree where formulas for the roots are either very complex or not possible to find.
Type: Problem-Solving Task
This task asks students to write expressions for various problems involving distance per units of volume.
Type: Problem-Solving Task
For a polynomial function p, a real number r is a root of p if and only if p(x) is evenly divisible by x-r. This fact leads to one of the important properties of polynomial functions: a polynomial of degree d can have at most d roots. This is the first of a sequence of problems aiming at showing this fact. The teacher should pay close attention to the logic used in the solution to part (c) where the divisibility of ax2+bx+c by x-r is obtained not by performing long division but by using the result of long division of these polynomials; namely, that said division will result in an expression of the following form:
ax2+bx+c=(x-r)l(x)+k
where l is a linear polynomial and k is a number.
This task could be used either for assessment or for instructional purposes. If it is used for assessment, parts (a) and (b) are more suitable than part (c). Each of the questions in this task could be formulated as an if and only if statement but the other implication, namely that f(x) is divisible by x-r if and only if r is a root of f. The direction not presented in this task is more straightforward and so has been left out.
Type: Problem-Solving Task
The context of this example is the spread of a flu virus on campus and the related sale of tissue boxes sold. Students interpret the composite function and determine values simply by using the tables of values.
Type: Problem-Solving Task
In this task, students explore the inverse relationship between an exponential function and a logarithmic function. The task is to determine the relevant composite functions, their graphs, and the domain and range of each.
Type: Problem-Solving Task
In this example, fuel efficiency of a car can be analyzed by using rational expressions and operations with rational expressions.
Type: Problem-Solving Task
In this resource, a method of deriving the quadratic formula from a theoretical standpoint is demonstrated. This task is for instructional purposes only and builds on "Building an explicit quadratic function."
Type: Problem-Solving Task
This task develops reasoning behind the general formula for balances under continuously compounded interest. While this task itself specifically address the standard (F-BF), building functions from a context, an auxiliary purpose is to introduce and motivate the number e, which plays a significant role in the (F-LE) domain of tasks.
Type: Problem-Solving Task
This task compares the usefulness of different forms of a quadratic expression. Students have to choose which form most easily provides information about the maximum value, the zeros and the vertical intercept of a quadratic expression in the context of a real world situation. Rather than just manipulating one form into the other, students can make sense out of the structure of the expressions.
(From Algebra: Form and Function, McCallum et al., Wiley 2010)
Type: Problem-Solving Task
The purpose of this task is to help students see manipulation of expressions as an activity undertaken for a purpose.
Variation 1 of this task presents a related more complex expression already in the correct form to answer the question.
The expression arises in physics as the reciprocal of the combined resistance of two resistors in parallel. However, the context is not explicitly considered here.
Type: Problem-Solving Task
This task illustrates the process of rearranging the terms of an expression to reveal different aspects about the quantity it represents, precisely the language being used in standard MAFS.912.A-SSE.2.3. Students are provided with an expression giving the temperature of a container at a time t, and have to use simple inequalities (e.g., that 2t>0 for all t) to reduce the complexity of an expression to a form where bounds on the temperature of a container of ice cream are made apparent.
Type: Problem-Solving Task
Students explore and manipulate expressions based on the following statement:
A function f defined for -a < x="">< a="" is="" even="" if="" f(-x)="f(x)" and="" is="" odd="" if="" f(-x)="-f(x)" when="" -a="">< x="">< a.="" in="" this="" task="" we="" assume="" f="" is="" defined="" on="" such="" an="" interval,="" which="" might="" be="" the="" full="" real="" line="" (i.e.,="" a="">
Type: Problem-Solving Task
Students compare graphs of different quadratic functions, then produce equations of their own to satisfy given conditions.
This exploration can be done in class near the beginning of a unit on graphing parabolas. Students need to be familiar with intercepts, and need to know what the vertex is. It is effective after students have graphed parabolas in vertex form (y=a(x–h)2+k), but have not yet explored graphing other forms.
Type: Problem-Solving Task
This is a standard problem phrased in a non-standard way. Rather than asking students to perform an operation, expanding, it expects them to choose the operation for themselves in response to a question about structure. Students must understand the need to transform the factored form of the quadratic expression (a product of sums) into a sum of products in order to easily see a, the coefficient of the x2 term; k, the leading coefficient of the x term; and n, the constant term.
Type: Problem-Solving Task
Students are asked to interpret the effect on the value of an expression given a change in value of one of the variables.
Type: Problem-Solving Task
Students examine and answer questions related to a scenario similar to a "mixture" problem involving two different mixtures of fertilizer. In this example, students determine and then compare expressions that correspond to concentrations of various mixtures. Ultimately, students generalize the problem and verify conclusions using algebraic rather than numerical expressions.
Type: Problem-Solving Task
Students are asked to interpret expressions and equations within the context of the amounts of caramels and truffles in a box of candy.
Type: Problem-Solving Task
This problem asks students to consider algebraic expressions calculating the number of floor tiles in given patterns. The purpose of this task is to give students practice in reading, analyzing, and constructing algebraic expressions, attending to the relationship between the form of an expression and the context from which it arises. The context here is intentionally thin; the point is not to provide a practical application to kitchen floors, but to give a framework that imbues the expressions with an external meaning.
Type: Problem-Solving Task
The goal of this task is to develop an understanding of rational exponents (MAFS.912.N-RN.1.1); however, it also raises important issues about distinguishing between linear and exponential behavior (MAFS.912.F-LE.1.1c) and it requires students to create an equation to model a context (MAFS.912.A-CED.1.2).
Type: Problem-Solving Task
This resource describes a simple scenario which can be represented by the use of variables. Students are asked to examine several variable expressions, interpret their meaning, and describe what quantities they each represent in the given context.
Type: Problem-Solving Task
In this task students interpret the relative size of variable expressions involving two variables in the context of a real world situation. All given expressions can be interpreted as quantities that one might study when looking at two animal populations.
Type: Problem-Solving Task
This resource involves simplifying algebraic expressions that involve complex numbers and various algebraic operations.
Type: Problem-Solving Task
The purpose of this task is to identify the structure in the two algebraic expressions by interpreting them in terms of a geometric context. Students will have likely seen this type of process before, so the principal source of challenge in this task is to encourage a multitude and variety of approaches, both in terms of the geometric argument and in terms of the algebraic manipulation.
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
This task requires students to recognize the graphs of different (positive) powers of x.
Type: Problem-Solving Task
This problem solving task examines, in a graphical setting, the impact of adding a scalar, multiplying by a scalar, and making a linear substitution of variables on the graph of the function f. This resource also includes standards alignment commentary and annotated solutions.
Type: Problem-Solving Task
Unit conversion problems provide a rich source of examples both for composition of functions (when several successive conversions are required) and inverses (units can always be converted in either of two directions).
Type: Problem-Solving Task
This task asks students to determine a recursive process from a context. Students who study computer programming will make regular use of recursive processes.
Type: Problem-Solving Task
This task asks students to use proportions of mass and volume to create ideal brine for saltwater fish tanks. It also asks students to compare graphs.
Type: Problem-Solving Task
This problem is a quadratic function example. The other tasks in this set illustrate MAFS.912.F.BF.1.1.a in the context of linear, exponential, and rational functions.
Type: Problem-Solving Task
In this task, students are asked to analyze a function and its inverse when the function is given as a table of values. In addition to finding values of the inverse function from the table, they also have to explain why the given function is invertible.
Type: Problem-Solving Task
The task addresses the first part of standard MAFS.912.F-BF.2.3: "Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative)."
Type: Problem-Solving Task
This task is designed to make students think about the meaning of the quantities presented in the context and choose which ones are appropriate for the two different constraints presented. In particular, note that the purpose of the task is to have students generate the constraint equations for each part (though the problem statements avoid using this particular terminology), and not to have students solve said equations. If desired, instructors could also use this task to touch on such solutions by finding and interpreting solutions to the system of equations created in parts (a) and (b).
Type: Problem-Solving Task
The purpose of this task is to introduce students to exponential growth. While the context presents a classic example of exponential growth, it approaches it from a non-standard point of view.
Type: Problem-Solving Task
In the middle grades, students have lots of experience analyzing and comparing linear functions using graphs, table, symbolic expressions, and verbal descriptions. In this task, students may choose a representation that suits them and then reason from within that representation.
Type: Problem-Solving Task
This task illustrates several components of standard MAFS.912.F-BF.2.4.c: Find Inverse Functions. Here, instead of presenting two functions and asking the students to decide which on is invertible, students are asked to complete a table of input-output pairs for the functions in such a way that one of the functions is invertible and the other one is not.
Type: Problem-Solving Task
The primary purpose of this task is to lead students to a numerical and graphical understanding of the behavior of a rational function near a vertical asymptote, in terms of the expression defining the function.
Type: Problem-Solving Task
The purpose of this task is to give students practice constructing functions that represent a quantity of interest in a context, and then interpreting features of the function in the light of the context. It can be used as either an assessment or a teaching task.
Type: Problem-Solving Task
This task asks students to determine whether a the set of given functions is odd, even, or neither.
Type: Problem-Solving Task
The purpose of this task is to emphasize the use of the Remainder Theorem (a discussion of which should obviously be considered as a prerequisite for the task) as a method for determining structure in polynomial in equations, and in this particular instance, as a replacement for division of polynomials.
Type: Problem-Solving Task
Student Center Activity
This video will demonstrate how to multiply polynomials.
Type: Student Center Activity
Tutorials
This video demonstrates how to multiply complex numbers using distributive property and FOIL method.
Type: Tutorial
You will learn in this video how to solve Quadratic Equations using the Quadratic Formula.
Type: Tutorial
You will learn how the parent function for a quadratic function is affected when f(x) = x2.
Type: Tutorial
You will learn int his video how to solve the Quadratic Equation by Completing the Square.
Type: Tutorial
This video will demonstrate how to simplify square roots involving variables.
Type: Tutorial
This tutorial gives an introduction to the unit circle. It also extends the students knowledge of SOH CAH TOA so that they can define trigonometric functions for a broader class of angles.
Type: Tutorial
This video is an example of solving a system of linear equations by elimination where the system has infinite solutions.
Type: Tutorial
This video shows how to solve a system of equations through simple elimination.
Type: Tutorial
This video explains how to identify systems of equations without a solution.
Type: Tutorial
This video will demonstrate how to solve radical equations with additional practice problems.
Type: Tutorial
This video will demonstrate how to subtract complex numbers.
Type: Tutorial
This video tutorial shows students: the standard form of a polynomial, how to identify polynomials, how to determine the degree of a polynomial, how to add and subtract polynomials, and how to represent the area of a shape as an addition or subtraction of polynomials.
Type: Tutorial
This video shows how to solve systems of equations by elimination.
Type: Tutorial
In this video tutorial students will learn how to solve quadratic equations by square roots.
Type: Tutorial
This video is an introduction to the elimination method of solving a system of equations.
Type: Tutorial
This video gives an introduction to 'i' and imaginary numbers. From this tutorial, students will learn the rules of imaginary numbers.
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
This video demonstrates solving a word problem by creating a system of linear equations that represents the situation and solving them using elimination.
Type: Tutorial
In this tutorial, students will learn how to solve and graph a system of equations.
Type: Tutorial
This tutorial shows students how to solve a system of linear equations by graphing the two equations on the same coordinate plane and identifying the intersection point.
Type: Tutorial
This tutorial shows how to solve a system of equations by graphing. Students will see what a no solution system of equations looks like in a graph.
Type: Tutorial
This tutorial shows how to solve a system of equations using substitution.
Type: Tutorial
This resource discusses dividing a polynomial by a monomial and also dividing a polynomial by a polynomial using long division.
Type: Tutorial
This video explains how to subtract polynomials with multiple variables and reinforces how to distribute a negative number.
Type: Tutorial
This video covers squaring a binomial with two variables. Students will be given the area of a square.
Type: Tutorial
This 8 minute video will show step-by-step directions for using the elimination method to solve a system of linear equations.
Type: Tutorial
This video provides a real-world scenario and step-by-step instructions to constructing equations using two variables. Possible follow-up videos include Plotting System of Equations - Yoga Plan, Solving System of Equations with Substitution - Yoga Plan, and Solving System of Equations with Elimination - Yoga Plan.
Type: Tutorial
Finding the 4th term in recursively defined sequence
Type: Tutorial
This tutorial will help the students to identify the vertex of a parabola from the equation, and then graph the parabola.
Type: Tutorial
This tutorial helps the learners to graph the equation of a quadratic function using the coordinates of the vertex of a parabola and its x- intercepts.
Type: Tutorial
Evaluating Expressions with Two Variables
Type: Tutorial
This tutorial will help you to learn about exponential functions by graphing various equations representing exponential growth and decay.
Type: Tutorial
Learn how to evaluate an expression with variables using a technique called substitution (or "plugging in").
Type: Tutorial
Our focus here is understanding that a variable is just a letter or symbol (usually a lower case letter) that can represent different values in an expression. We got this. Just watch.
Type: Tutorial
This tutorial demonstrates how to use the power of a power property with both numerals and variables.
Type: Tutorial
This lecture shows how algebra is used to solve problems involving mixtures of solutions of different concentrations.
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
Systems of two linear equations in two variables can have a single solution, no solutions, or an infinite number of solutions. This video gives a great description of inconsistent, dependent, and independent systems. A consistent independent system of equations will have one solution. A consistent dependent system of equations will have infinite number of solutions, and an inconsistent system of equations will have no solution. This tutorial also provides information on how to distinguish a given system of linear equations as inconsistent, independent, or dependent system by looking at the slope and intercept.
Type: Tutorial
Systems of two equations in x and y can be solved by adding the equations to create a new equation with one variable eliminated. This new equation can then be solved to find the value of the remaining variable. That value is then substituted into either equation to find the value of other variable.
Type: Tutorial
A system of two equations in x and y can be solved by rearranging one equation to represent x in terms of y, and "substituting" this expression for x in the other equation. This creates an equation with only y which can then be solved to find y's value. This value can then be substituted into either equation to find the value of x.
Type: Tutorial
The video tutorial discusses about two typical polynomial multiplications. First, squaring a binomial and second, product of a sum and difference.
Type: Tutorial
Binomials are the polynomials with two terms. This tutorial will help the students learn about the multiplication of binomials. In multiplication, we need to make sure that each term in the first set of parenthesis multiplies each term in the second set.
Type: Tutorial
This resource explores the electromagnetic spectrum and waves by allowing the learner to observe the refraction of light as it passes from one medium to another, study the relation between refraction of light and the refractive index of the medium, select from a list of materials with different refractive indicecs, and change the light beam from white to monochromatic and observe the difference.
Type: Tutorial
- Observe how the eye's muscles change the shape of the lens in accordance with the distance to the object being viewed
- Indicate the parts of the eye that are responsible for vision
- View how images are formed in the eye
Type: Tutorial
- Learn how a concave spherical mirror generates an image
- Observe how the size and position of the image changes with the object distance from the mirror
- Learn the difference between a real image and a virtual image
- Learn some applications of concave mirrors
Type: Tutorial
- Learn how a convex mirror forms the image of an object
- Understand why convex mirrors form small virtual images
- Observe the change in size and position of the image with the change in object's distance from the mirror
- Learn some practical applications of convex mirrors
Type: Tutorial
- Observe the change of color of a black body radiator upon changes in temperature
- Understand that at 0 Kelvin or Absolute Zero there is no molecular motion
Type: Tutorial
This resource explains how a solar cell converts light energy into electrical energy. The user will also learn about the different components of the solar cell and observe the relationship between photon intensity and the amount of electrical energy produced.
Type: Tutorial
- Observe that light is composed of oscillating electric and magnetic waves
- Explore the propagation of an electromagnetic wave through its electric and magnetic field vectors
- Observe the difference in propagation of light of different wavelengths
Type: Tutorial
- Explore the relationship between wavelength, frequency, amplitude and energy of an electromagnetic wave
- Compare the characteristics of waves of different wavelengths
Type: Tutorial
- Learn to trace the path of propagating light waves using geometrical optics
- Observe the effect of changing parameters such as focal length, object dimensions and position on image properties
- Learn the equations used in determining the size and locations of images formed by thin lenses
Type: Tutorial
Video/Audio/Animations
With an often unexpected outcome from a simple experiment, students can discover the factors that cause and influence thermohaline circulation in our oceans. In two 45-minute class periods, students complete activities where they observe the melting of ice cubes in saltwater and freshwater, using basic materials: clear plastic cups, ice cubes, water, salt, food coloring, and thermometers. There are no prerequisites for this lesson but it is helpful if students are familiar with the concepts of density and buoyancy as well as the salinity of seawater. It is also helpful if students understand that dissolving salt in water will lower the freezing point of water. There are additional follow up investigations that help students appreciate and understand the importance of the ocean's influence on Earth's climate.
Type: Video/Audio/Animation
This video will demonstrate how to solve a quadratic equation using square roots.
Type: Video/Audio/Animation
This video demonstrates how to determine if a relation is a function and how to identify the domain.
Type: Video/Audio/Animation
Exponents are not only integers. They can also be fractions. Using the rules of exponents, we can see why a number raised to the power " one over n" is equivalent to the nth root of that number.
Type: Video/Audio/Animation
Exponents are not only integers and unit fractions. An exponent can be any rational number expressed as the quotient of two integers.
Type: Video/Audio/Animation
Radical expressions can often be simplified by moving factors which are perfect roots out from under the radical sign.
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
When should a system of equations with multiple variables be used to solve an Algebra problem, instead of using a single equation with a single variable?
Type: Video/Audio/Animation
The points of intersection of two graphs represent common solutions to both equations. Finding these intersection points is an important tool in analyzing physical and mathematical systems.
Type: Video/Audio/Animation
This chapter presents a new look at the logic behind adding equations- the essential technique used when solving systems of equations by elimination.
Type: Video/Audio/Animation
The point-slope form of the equation for a line can describe any non-vertical line in the Cartesian plane, given the slope and the coordinates of a single point which lies on the line.
Type: Video/Audio/Animation
The two point form of the equation for a line can describe any non-vertical line in the Cartesian plane, given the coordinates of two points which lie on the line.
Type: Video/Audio/Animation
Literal equations are formulas for calculating the value of one unknown quantity from one or more known quantities. Variables in the formula are replaced by the actual or 'literal' values corresponding to a specific instance of the relationship.
Type: Video/Audio/Animation
This video takes a look at rearranging a formula to highlight a quantity of interest.
Type: Video/Audio/Animation
This video demonstrates writing a function that represents a real-life scenario.
Type: Video/Audio/Animation
This video gives a more in-depth look at graphing quadratic functions than previously offered in Quadratic Functions 1.
Type: Video/Audio/Animation
This learning video introduces students to the world of Fractal Geometry through the use of difference equations. As a prerequisite to this lesson, students would need two years of high school algebra (comfort with single variable equations) and motivation to learn basic complex arithmetic. Ms. Zager has included a complete introductory tutorial on complex arithmetic with homework assignments downloadable here. Also downloadable are some supplemental challenge problems. Time required to complete the core lesson is approximately one hour, and materials needed include a blackboard/whiteboard as well as space for students to work in small groups. During the in-class portions of this interactive lesson, students will brainstorm on the outcome of the chaos game and practice calculating trajectories of difference equations.
Type: Video/Audio/Animation
Khan Academy video tutorial on graphing linear equations: "Algebra: Graphing Lines 1"
Type: Video/Audio/Animation
This Khan Academy video tutorial introduces averages and algebra problems involving averages.
Type: Video/Audio/Animation
Virtual Manipulatives
This resource will assess students' understanding of addition and subtraction of polynomials.
Type: Virtual Manipulative
This resource can be used to assess students' understanding of solving quadratic equation by taking the square root. A great resource to view prior to this is "Solving quadratic equations by square root' by Khan Academy.
Type: Virtual Manipulative
In this activity, students adjust slider bars which adjust the coefficients and constants of a linear function and examine how their changes affect the graph. The equation of the line can be in slope-intercept form or standard form. This activity allows students to explore linear equations, slopes, and y-intercepts and their visual representation on a graph. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
Type: Virtual Manipulative
In this online activity, students burn a simulated forest and adjust the probability that the fire spreads from one tree to the other. This simulation also records data for each trial including the burn probability, where the fire started, the percent of trees burned, and how long the fire lasted. This activity allows students to explore the idea of chaos in a simulation of a realistic scenario. 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 are linked to the applet.
Type: Virtual Manipulative
In this activity, students select one of three doors in an attempt to find a prize that is hidden behind one of them. After their first selection, one of the doors that doesn't have the prize behind it is revealed and the student has to decide whether to switch to the one remaining door or stay on the door of their first choice. This situation, referred to as the Monty Hall problem, was made famous on the show "Let's Make A Deal" with host Monty Hall. This activity allows students to explore the idea of conditional probability as well as unexpected probability. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
Type: Virtual Manipulative
In this activity, students adjust how many sections there are on a fair spinner then run simulated trials on that spinner as a way to develop concepts of probability. A table next to the spinner displays the theoretical probability for each color section of the spinner and records the experimental probability from the spinning trials. This activity allows students to explore the topics of experimental and theoretical probability by seeing them displayed side by side for the spinner they have created. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
Type: Virtual Manipulative
This resource provides linear functions in standard form and asks the user to graph it using intercepts on an interactive graph below the problem. Immediate feedback is provided, and for incorrect responses, each step of the solution is thoroughly modeled.
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
This is a graphing tool/activity for students to deepen their understanding of polynomial functions and their corresponding graphs. This tool is to be used in conjunction with a full lesson on graphing polynomial functions; it can be used either before an in depth lesson to prompt students to make inferences and connections between the coefficients in polynomial functions and their corresponding graphs, or as a practice tool after a lesson in graphing the polynomial functions.
Type: Virtual Manipulative
With this online tool, students adjust the standard deviation and sample size of a normal distribution to see how it will affect a histogram of that distribution. This activity allows students to explore the effect of changing the sample size in an experiment and the effect of changing the standard deviation of a normal distribution. Tabs at the top of the page provide access to supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
Type: Virtual Manipulative
In this online tool, students input a function to create a graph where the constants, coefficients, and exponents can be adjusted by slider bars. This tool allows students to explore graphs of functions and how adjusting the numbers in the function affect the graph. Using tabs at the top of the page you can also access supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.
Type: Virtual Manipulative
This online manipulative allows the student to simulate placing marbles into a bag and finding the probability of pulling out certain combinations of marbles. This allows exploration of probabilities of multiple events as well as probability with and without replacement. The tabs above the applet provide access to supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.
Type: Virtual Manipulative
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 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 applet allows users to set up various geometric series with a visual representation of the successive terms, and the corresponding sum of those terms.
Type: Virtual Manipulative
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