# Liberal Arts Mathematics 2   (#1207310)

## 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:

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

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

### General Information

Course Number: 1207310
Abbreviated Title: LIB ARTS MATH 2
Number of Credits: One (1) credit
Course Length: Year (Y)
Course Attributes:
• Highly Qualified Teacher (HQT) Required
• Florida Standards Course
Course Level: 2
Course Status: Course Approved

## Educator Certifications

One of these educator certification options is required to teach this course.

## Student Resources

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

## Original Student Tutorials

Solving Rational Equations: Cross Multiplying:

Learn how to solve rational linear and quadratic equations using cross multiplication in this interactive tutorial.

Type: Original Student Tutorial

The Year-Round School Debate: Identifying Faulty Reasoning â€” Part Two:

Practice identifying 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 Part One before Part Two! Click HERE to launch Part One.

Type: Original Student Tutorial

The Year-Round School Debate: Identifying Faulty Reasoning â€“ Part One:

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

Evaluating an Argument â€“ Part Four: JFKâ€™s Inaugural Address:

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.

• Click HERE to launch Part One.
• Click HERE to launch Part Two.
• Click HERE to launch Part Three.

Type: Original Student Tutorial

Evaluating an Argument â€“ Part Three: JFKâ€™s Inaugural Address:

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!

• Click HERE to launch Part One.
• Click HERE to launch Part Two.

Type: Original Student Tutorial

Where IS that Cell Tower?:

Find the location and coverage area of cell towers by completing the square to determine the center and radius of a circle given its equation in this interactive tutorial.

Type: Original Student Tutorial

Circle Up!:

Learn how to write the equation of a circle using Pythagorean Theorem given its center and radius using step-by-step instructions on this interactive tutorial.

Type: Original Student Tutorial

Finding the Zeros of Quadratic Functions:

Quadratic functions can be used to model real-world phenomena. Key features of quadratic functions such as maximum values and zeros can often reveal important qualities of these phenomena. By the end of this tutorial, you should be able to find the zeros of a quadratic function and interpret their meaning in real-world contexts.

Type: Original Student Tutorial

Ready for Takeoff! -- Part Two:

Want to learn about Amelia Earhart, one of the most famous female aviators of all time? If so, then this interactive tutorial is for YOU! This tutorial is Part Two of a two-part series. In this series, you will study a speech by Amelia Earhart. You will practice identifying the purpose of her speech and practice identifying her use of rhetorical appeals (ethos, logos, pathos, Kairos). You will also evaluate the effectiveness of Earhart's rhetorical choices based on the purpose of her speech.

Please complete Part One before beginning Part Two. Click HERE to view Part One.

Type: Original Student Tutorial

Ready for Takeoff! -- Part One:

Want to learn about Amelia Earhart, one of the most famous female aviators of all time? If so, then this interactive tutorial is for YOU! This tutorial is Part One of a two-part series. In this series, you will study a speech by Amelia Earhart. You will practice identifying the purpose of her speech and practice identifying her use of rhetorical appeals (ethos, logos, pathos, Kairos). You will also evaluate the effectiveness of Earhart's rhetorical choices based on the purpose of her speech.

Type: Original Student Tutorial

Untangling Food Webs:

Learn how living organisms can be organized into food webs and how energy is transferred through a food web from producers to consumers to decomposers. This interactive tutorial also includes interactive knowledge checks.

Type: Original Student Tutorial

Long Division With Polynomials:

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.

Type: Original Student Tutorial

Finding the Maximum or Minimum of a Quadratic Function:

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

Comparing Mitosis and Meiosis:

Compare and contrast mitosis and meiosis in this interactive tutorial. You'll also relate them to the processes of sexual and asexual reproduction and their consequences for genetic variation.

Type: Original Student Tutorial

Creating Exponential Functions:

By the end of this tutorial, you should be able construct an exponential function from a graph, from a table of values, and from a description of a relationship in the real world.

Type: Original Student Tutorial

The graph of a quadratic equation is called a parabola [puh-ra-bow-luh]. The key features we will focus on in this tutorial are the vertex (a maximum or minimum extreme) and the direction of its opening. You will learn how to examine a quadratic equation written in vertex form in order to distinguish each of these key features.

Type: Original Student Tutorial

Cancer: Mutated Cells Gone Wild!:

Explore the relationship between mutations, the cell cycle, and uncontrolled cell growth which may result in cancer with this interactive tutorial.

Type: Original Student Tutorial

Climbing Around the Hominin Family Tree:

Learn to identify basic trends in the evolutionary history of humans, including walking upright, brain size, jaw size, and tool use in "Climbing Around the Hominin Family Tree" online tutorial.

Type: Original Student Tutorial

## Educational Games

Rewriting Expressions: Simplifying Rational Expressions With Exponents:

In this challenge game, you will be simplifying fractional expressions with exponents. Use the "Teach Me" button to review content before the challenge. During the challenge you get one free solve and two hints! After the challenge, review the problems as needed. Try again to get all challenge questions right! Question sets vary with each game, so feel free to play the game multiple times as needed! Good luck!

Type: Educational Game

Timed Algebra Quiz:

In this timed activity, students solve linear equations (one- and two-step) or quadratic equations of varying difficulty depending on the initial conditions they select. This activity allows students to practice solving equations while the activity records their score, so they can track their progress. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Educational Game

Algebra Four:

In this activity, two students play a simulated game of Connect Four, but in order to place a piece on the board, they must correctly solve an algebraic equation. This activity allows students to practice solving equations of varying difficulty: one-step, two-step, or quadratic equations and using the distributive property if desired. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the Java applet.

Type: Educational Game

## 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

2. Input joint frequency data.
3. Click the second tab at the bottom of the window to see the automatic calculations.

Type: Educational Software / Tool

## Perspectives Video: Experts

Type: Perspectives Video: Expert

Mathematically Exploring the Wakulla Caves:

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

Type: Perspectives Video: Expert

MicroGravity Sensors & Statistics:

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

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

Making Candy: Illuminating Exponential Growth:

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

Population Sampling and Beekeeping:

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

Rain and Lightning:

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.

Breakfast Before School:

The purpose of this task is to assess a student's ability to explain the meaning of independence in a simple context.

Musical Preferences:

This problem solving task asks students to make deductions about what kind of music students like by examining a table with data.

Students are asked to choose the best sampling method for choosing the new School Advisory Panel.

Words and Music II:

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.

The Titanic 2:

This task lets students explore the concepts of probability as a fraction of outcomes and using two-way tables of data.

The Titanic 1:

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

The Titanic 3:

This problem solving task asks students to determine probabilities and draw conclusions about the survival rates on the Titanic by consulting a table of data.

Algae Blooms:

In this example, students are asked to write a function describing the population growth of algae. It is implied that this is exponential growth.

A Midpoint Miracle:

This problem solving task gives students the opportunity to prove a fact about quadrilaterals: that if we join the midpoints of an arbitrary quadrilateral to form a new quadrilateral, then the new quadrilateral is a parallelogram, even if the original quadrilateral was not.

Slopes and Circles:

The purpose of this task is to lead students through an algebraic approach to a well-known result from classical geometry, namely, that a point X is on the circle of diameter AB whenever ?AXB is a right angle

Unit Squares and Triangles:

This problem solving task asks students to find the area of a triangle by using unit squares and line segments.

Triangles inscribed in a circle:

This problem solving task challenges students to use ideas about linear functions in order to determine when certain angles are right angles.

What functions do two graph points determine?:

This problem solving task challenges students to find the linear, exponential and quadratic functions based on two points.

US Population 1982-1988:

This problem solving task asks students to predict and model US population based on a chart of US population data from 1982 to 1988.

US Population 1790-1860:

This problem solving task asks students to solve five exponential and linear function problems based on a US population chart for the years 1790-1860.

Two Points Determine an Exponential Function II:

This problem solving tasks asks students to find the values of points on a graph.

Two Points Determine an Exponential Function I:

This problem solving task asks students to graph a function and find the values of points on a graph.

Taxi!:

This simple conceptual problem does not require algebraic manipulation, but requires students to articulate the reasoning behind each statement.

Snail Invasion:

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.

Sandia Aerial Tram:

The task provides an opportunity for students to engage in detailed analysis of the rate of change of the elevation.

Rumors:

This problem is an exponential function example that uses the real-world problem of how fast rumors spread.

Rising Gas Prices - Compounding and Inflation:

The purpose of this task is to give students an opportunity to explore various aspects of exponential models (e.g., distinguishing between constant absolute growth and constant relative growth, solving equations using logarithms, applying compound interest formulas) in the context of a real world problem with ties to developing financial literacy skills. In particular, students are introduced to the idea of inflation of prices of a single commodity, and are given a very brief introduction to the notion of the Consumer Price Index for measuring inflation of a body of goods.

Newton's Law of Cooling:

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.

Linear or exponential?:

This task gives a variation of real-life contexts which could be modeled by a linear or exponential function. The key distinguishing feature between the two is whether the change by equal factors over equal intervals (exponential functions), or by a constant increase per unit interval (linear functions). The task could either be used as an assessment problem on this distinction, or used as an introduction to the differences between these very important classes of functions.

Linear Functions:

This task requires students to use the fact that on the graph of the linear function h(x) = ax + b, the y-coordinate increases by a when x increases by one. Specific values for a and b were left out intentionally to encourage students to use the above fact as opposed to computing the point of intersection, (p,q), and then computing respective function values to answer the question.

The Circle and The Line:

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.

Do two points always determine an exponential function?:

This problem complements the problem "Do two points always determine a linear function?'' There are two constraints on a pair of points R1 and R2 if there is an exponential function f(x) = ae^bx whose graph contains R1 and R2.

Do two points always determine a linear function?:

This problem complements the problem "Do two points always determine a linear function?'' There are two constraints on a pair of points R1 and R2 if there is an exponential function f(x) = ae^bx whose graph contains R1 and R2. First, the y-coordinates of R1 and R2 cannot have different signs, that is it cannot be that one is positive while the other is negative. This is because the function g(x) = ex takes only positive values. Consequently, f(x) = ae^bx cannot take both positive and negative values. Furthermore, the only way aebx can be zero is if a = 0 and then the function is linear rather than exponential. As long as the y-coordinates of R1 and R2 are non-zero and have the same sign, there is a unique exponential function f(x) = ae^bx whose graph contains R1 and R2.

Comparing Exponentials:

This task gives students an opportunity to work with exponential functions in a real world context involving continuously compounded interest. They will study how the base of the exponential function impacts its growth rate and use logarithms to solve exponential equations.

Carbon 14 Dating, Variation 2:

This exploratory task requires the student to use properties of exponential functions in order to estimate how much Carbon 14 remains in a preserved plant after different amounts of time.

Carbon 14 Dating in Practice II:

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.

Carbon 14 Dating in Practice I:

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.

Carbon 14 Dating:

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.

This task involves a fairly straightforward decaying exponential. Filling out the table and developing the general formula is complicated only by the need to work with a fraction that requires decisions about rounding and precision.

Bacteria Populations:

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.

A Saturating Exponential:

This task provides an interesting context to ask students to estimate values in an exponential function using a graph.

Population and Food Supply:

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).

Braking Distance:

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.

This task asks students to consider the linear and quadratic functions shown on a graph, and use quadratic functions to find the coordinates.

A Cubic Identity:

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.

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).

Two Squares are Equal:

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.

Exponential growth versus linear growth I:

The purpose of this task it to have students discover how (and how quickly) an exponentially increasing quantity eventually surpasses a linearly increasing quantity. Students' intuitions will probably have them favoring Option A for much longer than is actually the case, especially if they are new to the phenomenon of exponential growth. Teachers might use this surprise as leverage to segue into a more involved task comparing linear and exponential growth.

Exponential Functions:

This task requires students to use the fact that the value of an exponential function f(x) = a · b^x increases by a multiplicative factor of b when x increases by one. It intentionally omits specific values for c and d in order to encourage students to use this fact instead of computing the point of intersection, (p,q), and then computing function values to answer the question.

Equal Factors over Equal Intervals:

This problem assumes that students are familiar with the notation x0 and Δx. However, the language "successive quotient" may be new.

Equal Differences over Equal Intervals 2:

This task assumes that students are familiar with the ?x and ?y notations. Students most likely developed this familiarity in their work with slope.

Equal Differences over Equal Intervals 1:

An important property of linear functions is that they grow by equal differences over equal intervals. In F.LE Equal Differences over Equal Intervals 1, students prove this for equal intervals of length one unit, and note that in this case the equal differences have the same value as the slope.

In the Billions and Linear Modeling:

This problem-solving task asks students to examine if linear modeling would be appropriate to describe and predict population growth from select years.

In the Billions and Exponential Modeling:

This problem-solving task provides students an opportunity to experiment with modeling real data by using population growth rates from the past two centuries.

Interesting Interest Rates:

This problem-solving task challenges students to write expressions and create a table to calculate how much money can be gained after investing at different banks with different interest rates.

Illegal Fish:

This problem-solving task asks students to describe exponential growth through a real-world problem involving the illegal introduction of fish into a lake.

This task asks students to write equations to predict how much money will be in a savings account at the end of each year, based on different factors like interest rates.

Identifying Functions:

This problem-solving emphasizes the expectation that students know linear functions grow by constant differences over equal intervals and exponential functions grow by constant factors over equal intervals.

Finding Parabolas through Two Points:

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.

Exponential growth versus polynomial growth:

This problem solving task shows that an exponential function takes larger values than a cubic polynomial function provided the input is sufficiently large. This resource also includes standards alignment commentary and annotated solutions.

Exponential growth versus linear growth II:

This task asks students to calculate exponential functions with a base larger than one.

Which Function?:

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.

Throwing Baseballs:

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.

Springboard Dive:

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.

Trina's Triangles:

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.

Zeroes and factorization of a general polynomial:

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.

Zeroes and factorization of a non polynomial function:

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.

Checking a Calculation of a Decimal Exponent:

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.

Triangle Series:

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.

Course of Antibiotics:

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.

Cantor Set:

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.

Forms of Exponential Expressions:

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.

Calculating the Square Root of 2:

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

Zeroes and factorization of a quadratic polynomial II:

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.

Zeroes and factorization of a quadratic polynomial I:

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.

Combined Fuel Efficiency:

In this example, fuel efficiency of a car can be analyzed by using rational expressions and operations with rational expressions.

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."

Profit of a Company:

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)

Increasing or Decreasing? Variation 2:

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.

Ice Cream:

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.

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.

Equivalent Expressions:

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.

Extending the Definitions of Exponents, Variation 2:

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).

Animal Populations:

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.

Computations with Complex Numbers:

This resource involves simplifying algebraic expressions that involve complex numbers and various algebraic operations.

Operations with Rational and Irrational Numbers:

This task has students experiment with the operations of addition and multiplication, as they relate to the notions of rationality and irrationality.

Seeing Dots:

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.

Graphs of Power Functions:

This task requires students to recognize the graphs of different (positive) powers of x.

Kimi and Jordan:

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.

The Missing Coefficient:

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.

## Tutorials

Multiplying Complex Numbers:

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

Graphs and Solutions of Functions in Quadratic Equations:

You will learn how the parent function for a quadratic function is affected when f(x) = x2.

Type: Tutorial

Learning How to Complete the Square:

You will learn int his video how to solve the Quadratic Equation by Completing the Square.

Type: Tutorial

Graphs of second degree polynomials:

In this tutorial, students will look at input and output values of quadratic functions to help them understand why the graph of a second degree polynomial curves.

Type: Tutorial

How to Subtract Complex Numbers:

This video will demonstrate how to subtract complex numbers.

Type: Tutorial

This video will demonstrate how to add complex numbers.

Type: Tutorial

Solving Quadratic Equations by Square Roots:

In this video tutorial students will learn how to solve quadratic equations by square roots.

Type: Tutorial

Introduction to i and imaginary numbers:

This video gives an introduction to 'i' and imaginary numbers. From this tutorial, students will learn the rules of imaginary numbers.

Type: Tutorial

Division of Polynomials:

This resource discusses dividing a polynomial by a monomial and also dividing a polynomial by a polynomial using long division.

Type: Tutorial

Geometric series:

Geometric series

Type: Tutorial

Graphing Quadractic Functions in Vertex Form:

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 will help the learners to graph the equation of the quadratic function using the coordinates of the vertex of a parabola adn its x- intercepts.

Type: Tutorial

Graphing Exponential Equations:

This tutorial will help you to learn about the exponential functions by graphing various equations representing exponential growth and decay.

Type: Tutorial

Power of a Power Property:

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

Type: Tutorial

Parallel Lines:

Parallel lines have the same slope and no points in common. However, it is not always obvious whether two equations describe parallel lines or the same line.

Type: Tutorial

Perpendicular Lines:

Perpendicular lines have slopes which are negative reciprocals of each other, but why?

Type: Tutorial

Population Demographic Lab:

This lab simulation allows you to use real demographic data, collected by the US Census Bureau, to analyze and make predictions centered around demographic trends. You will explore factors that impact the birth, death and immigration rate of a population and learn how the population transitions having taken place globally.

Type: Tutorial

Dividing Polynomials:

This tutorial will help the learners practice division of polynomials. Students will recognize that dividing polynomials is similar to simplifying fractions.

Type: Tutorial

## Video/Audio/Animations

Solving Quadratic Equations using Square Roots:

This video will demonstrate how to solve a quadratic equation using square roots.

Type: Video/Audio/Animation

Roots and Unit Fraction Exponents:

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

Rational Exponents:

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

Parallel Lines 2:

This video shows how to determine which lines are parallel from a set of three different equations.

Type: Video/Audio/Animation

Parallel Lines:

This video illustrates how to determine if the graphs of a given set of equations are parallel.

Type: Video/Audio/Animation

Perpendicular Lines 2:

This video describes how to determine the equation of a line that is perpendicular to another line. All that is given initially the equation of a line and an ordered pair from the other line.

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

MIT BLOSSOMS - Fabulous Fractals and Difference Equations :

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

Graphing Lines 1:

Khan Academy video tutorial on graphing linear equations: "Algebra: Graphing Lines 1"

Type: Video/Audio/Animation

## Virtual Manipulatives

Solving Quadratics By Taking The Square Root:

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

Loan Calculator:

This virtual manipulative allows the user to explore scenarios of a loan repayment by manipulating the amount of the loan, interest rate, payment amount, frequency of payments, and length of the loan in years.

Type: Virtual Manipulative

Slope Slider:

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

Simple Monty Hall:

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

Spinner:

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

Linear Function Machine:

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

Type: Virtual Manipulative

Graphing Equations Using Intercepts:

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

Graphing Lines:

Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.

Type: Virtual Manipulative

Data Flyer:

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

Function Flyer:

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

Interactive Marbles:

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

Curve Fitting:

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

Type: Virtual Manipulative

Equation Grapher:

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

Fractal Tool:

Students investigate shapes that grow and change using an iterative process. Fractals are characterized by self-similarity, smaller sections that resemble the larger figure. From NCTM's Illuminations.

Type: Virtual Manipulative

Tool to Explore Exponential Functions:

This is an interactive applet in which students or teachers can visualize how changes in the parameters of the exponential function, y = a(b) x + c, affect the shape of the graph.

Type: Virtual Manipulative

Geometric and Harmonic Series- Limits:

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

## Parent Resources

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

## Perspectives Video: Professional/Enthusiasts

Making Candy: Illuminating Exponential Growth:

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

Population Sampling and Beekeeping:

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

Rain and Lightning:

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.

Breakfast Before School:

The purpose of this task is to assess a student's ability to explain the meaning of independence in a simple context.

Musical Preferences:

This problem solving task asks students to make deductions about what kind of music students like by examining a table with data.

Students are asked to choose the best sampling method for choosing the new School Advisory Panel.

Words and Music II:

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.

The Titanic 2:

This task lets students explore the concepts of probability as a fraction of outcomes and using two-way tables of data.

The Titanic 1:

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

The Titanic 3:

This problem solving task asks students to determine probabilities and draw conclusions about the survival rates on the Titanic by consulting a table of data.

Algae Blooms:

In this example, students are asked to write a function describing the population growth of algae. It is implied that this is exponential growth.

A Midpoint Miracle:

This problem solving task gives students the opportunity to prove a fact about quadrilaterals: that if we join the midpoints of an arbitrary quadrilateral to form a new quadrilateral, then the new quadrilateral is a parallelogram, even if the original quadrilateral was not.

Slopes and Circles:

The purpose of this task is to lead students through an algebraic approach to a well-known result from classical geometry, namely, that a point X is on the circle of diameter AB whenever ?AXB is a right angle

Unit Squares and Triangles:

This problem solving task asks students to find the area of a triangle by using unit squares and line segments.

Triangles inscribed in a circle:

This problem solving task challenges students to use ideas about linear functions in order to determine when certain angles are right angles.

What functions do two graph points determine?:

This problem solving task challenges students to find the linear, exponential and quadratic functions based on two points.

US Population 1982-1988:

This problem solving task asks students to predict and model US population based on a chart of US population data from 1982 to 1988.

US Population 1790-1860:

This problem solving task asks students to solve five exponential and linear function problems based on a US population chart for the years 1790-1860.

Two Points Determine an Exponential Function II:

This problem solving tasks asks students to find the values of points on a graph.

Two Points Determine an Exponential Function I:

This problem solving task asks students to graph a function and find the values of points on a graph.

Taxi!:

This simple conceptual problem does not require algebraic manipulation, but requires students to articulate the reasoning behind each statement.

Snail Invasion:

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.

Sandia Aerial Tram:

The task provides an opportunity for students to engage in detailed analysis of the rate of change of the elevation.

Rumors:

This problem is an exponential function example that uses the real-world problem of how fast rumors spread.

Rising Gas Prices - Compounding and Inflation:

The purpose of this task is to give students an opportunity to explore various aspects of exponential models (e.g., distinguishing between constant absolute growth and constant relative growth, solving equations using logarithms, applying compound interest formulas) in the context of a real world problem with ties to developing financial literacy skills. In particular, students are introduced to the idea of inflation of prices of a single commodity, and are given a very brief introduction to the notion of the Consumer Price Index for measuring inflation of a body of goods.

Newton's Law of Cooling:

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.

Linear or exponential?:

This task gives a variation of real-life contexts which could be modeled by a linear or exponential function. The key distinguishing feature between the two is whether the change by equal factors over equal intervals (exponential functions), or by a constant increase per unit interval (linear functions). The task could either be used as an assessment problem on this distinction, or used as an introduction to the differences between these very important classes of functions.

Linear Functions:

This task requires students to use the fact that on the graph of the linear function h(x) = ax + b, the y-coordinate increases by a when x increases by one. Specific values for a and b were left out intentionally to encourage students to use the above fact as opposed to computing the point of intersection, (p,q), and then computing respective function values to answer the question.

The Circle and The Line:

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.

Do two points always determine an exponential function?:

This problem complements the problem "Do two points always determine a linear function?'' There are two constraints on a pair of points R1 and R2 if there is an exponential function f(x) = ae^bx whose graph contains R1 and R2.

Do two points always determine a linear function?:

This problem complements the problem "Do two points always determine a linear function?'' There are two constraints on a pair of points R1 and R2 if there is an exponential function f(x) = ae^bx whose graph contains R1 and R2. First, the y-coordinates of R1 and R2 cannot have different signs, that is it cannot be that one is positive while the other is negative. This is because the function g(x) = ex takes only positive values. Consequently, f(x) = ae^bx cannot take both positive and negative values. Furthermore, the only way aebx can be zero is if a = 0 and then the function is linear rather than exponential. As long as the y-coordinates of R1 and R2 are non-zero and have the same sign, there is a unique exponential function f(x) = ae^bx whose graph contains R1 and R2.

Comparing Exponentials:

This task gives students an opportunity to work with exponential functions in a real world context involving continuously compounded interest. They will study how the base of the exponential function impacts its growth rate and use logarithms to solve exponential equations.

Carbon 14 Dating, Variation 2:

This exploratory task requires the student to use properties of exponential functions in order to estimate how much Carbon 14 remains in a preserved plant after different amounts of time.

Carbon 14 Dating in Practice II:

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.

Carbon 14 Dating in Practice I:

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.

Carbon 14 Dating:

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.

This task involves a fairly straightforward decaying exponential. Filling out the table and developing the general formula is complicated only by the need to work with a fraction that requires decisions about rounding and precision.

Bacteria Populations:

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.

A Saturating Exponential:

This task provides an interesting context to ask students to estimate values in an exponential function using a graph.

Population and Food Supply:

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).

Braking Distance:

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.

This task asks students to consider the linear and quadratic functions shown on a graph, and use quadratic functions to find the coordinates.

A Cubic Identity:

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.

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).

Two Squares are Equal:

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.

Exponential growth versus linear growth I:

The purpose of this task it to have students discover how (and how quickly) an exponentially increasing quantity eventually surpasses a linearly increasing quantity. Students' intuitions will probably have them favoring Option A for much longer than is actually the case, especially if they are new to the phenomenon of exponential growth. Teachers might use this surprise as leverage to segue into a more involved task comparing linear and exponential growth.

Exponential Functions:

This task requires students to use the fact that the value of an exponential function f(x) = a · b^x increases by a multiplicative factor of b when x increases by one. It intentionally omits specific values for c and d in order to encourage students to use this fact instead of computing the point of intersection, (p,q), and then computing function values to answer the question.

Equal Factors over Equal Intervals:

This problem assumes that students are familiar with the notation x0 and Δx. However, the language "successive quotient" may be new.

Equal Differences over Equal Intervals 2:

This task assumes that students are familiar with the ?x and ?y notations. Students most likely developed this familiarity in their work with slope.

Equal Differences over Equal Intervals 1:

An important property of linear functions is that they grow by equal differences over equal intervals. In F.LE Equal Differences over Equal Intervals 1, students prove this for equal intervals of length one unit, and note that in this case the equal differences have the same value as the slope.

In the Billions and Linear Modeling:

This problem-solving task asks students to examine if linear modeling would be appropriate to describe and predict population growth from select years.

In the Billions and Exponential Modeling:

This problem-solving task provides students an opportunity to experiment with modeling real data by using population growth rates from the past two centuries.

Interesting Interest Rates:

This problem-solving task challenges students to write expressions and create a table to calculate how much money can be gained after investing at different banks with different interest rates.

Illegal Fish:

This problem-solving task asks students to describe exponential growth through a real-world problem involving the illegal introduction of fish into a lake.

This task asks students to write equations to predict how much money will be in a savings account at the end of each year, based on different factors like interest rates.

Identifying Functions:

This problem-solving emphasizes the expectation that students know linear functions grow by constant differences over equal intervals and exponential functions grow by constant factors over equal intervals.

Finding Parabolas through Two Points:

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.

Exponential growth versus polynomial growth:

This problem solving task shows that an exponential function takes larger values than a cubic polynomial function provided the input is sufficiently large. This resource also includes standards alignment commentary and annotated solutions.

Exponential growth versus linear growth II:

This task asks students to calculate exponential functions with a base larger than one.

Which Function?:

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.

Throwing Baseballs:

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.

Springboard Dive:

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.

Trina's Triangles:

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.

Zeroes and factorization of a general polynomial:

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.

Zeroes and factorization of a non polynomial function:

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.

Checking a Calculation of a Decimal Exponent:

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.

Triangle Series:

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.

Course of Antibiotics:

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.

Cantor Set:

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.

Forms of Exponential Expressions:

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.

Calculating the Square Root of 2:

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

Zeroes and factorization of a quadratic polynomial II:

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.

Zeroes and factorization of a quadratic polynomial I:

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.

Combined Fuel Efficiency:

In this example, fuel efficiency of a car can be analyzed by using rational expressions and operations with rational expressions.

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."

Profit of a Company:

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)

Increasing or Decreasing? Variation 2:

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.

Ice Cream:

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.

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.

Equivalent Expressions:

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.

Extending the Definitions of Exponents, Variation 2:

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).

Animal Populations:

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.

Computations with Complex Numbers:

This resource involves simplifying algebraic expressions that involve complex numbers and various algebraic operations.

Operations with Rational and Irrational Numbers:

This task has students experiment with the operations of addition and multiplication, as they relate to the notions of rationality and irrationality.

Seeing Dots:

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.

Graphs of Power Functions:

This task requires students to recognize the graphs of different (positive) powers of x.

Kimi and Jordan:

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.

The Missing Coefficient:

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.

## Tutorials

Power of a Power Property:

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

Type: Tutorial

Dividing Polynomials:

This tutorial will help the learners practice division of polynomials. Students will recognize that dividing polynomials is similar to simplifying fractions.

Type: Tutorial

## Video/Audio/Animation

Graphing Lines 1:

Khan Academy video tutorial on graphing linear equations: "Algebra: Graphing Lines 1"

Type: Video/Audio/Animation

## Virtual Manipulatives

Solving Quadratics By Taking The Square Root:

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

Loan Calculator:

This virtual manipulative allows the user to explore scenarios of a loan repayment by manipulating the amount of the loan, interest rate, payment amount, frequency of payments, and length of the loan in years.

Type: Virtual Manipulative

Graphing Lines:

Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.

Type: Virtual Manipulative