In these activities, students explore Newton’s law of cooling by looking at the average rate of change of afunction that models a cooling situation. They examine the pattern in the average rates of change oversmall intervals and see that the average rates of change are proportional to the difference between theobject’s temperature and the ambient temperature. Students also check that a given model agrees withstated assumptions by computing and examining average rates of change. The notion that exponentialfactors can appear in other classes of functions, and reinforces the importance of the behavior ofexponential functions in mathematics and applied sciences is introduced. 

In these activities, students create a new function that measures the average rate of change of anexponential decay and half-life function and realize that the behavior of this new function is exponential.They explore a medical context in order to further understand the relationship between an exponentialfunction and its rate of change, and build on their understanding of the exponential behavior of theaverage rate of change function by looking at its outputs. Students study exponential functions of the form𝑦 = 𝐶𝑎𝑥, and connect information about the constants 𝐶 and 𝑎 to the shape of the graph of the function.Students also compare exponential functions using information about the base 𝑎. In addition tounderstanding that exponential growth corresponds to 𝑎 > 1 and that exponential decay corresponds to 0 <𝑎 < 1, students understand that as 𝑎 > 1 increases, the shape of the graph changes, and the graph becomessteeper for positive input values. Similarly, as 𝑎 decreases, for 0 < 𝑎 < 1, the shape of the graph changesand the function values go to zero more rapidly.

In these activities, students compute and interpret average rates of change (AROC) of polynomialfunctions in the context of examining the change in height while riding a rollercoaster. Students alsopractice algebraic manipulation of polynomials in the context of modeling the height of a drone above theground. The activities include a three-part activity which investigates a model for fuel consumption inwhich fuel consumption is represented as a function of speed, uses function composition to represent thefuel consumption as a function of time, and investigates total consumption as the area under theconsumption rate curve. 

In these activities, students recognized that exponential functions are either always increasing or alwaysdecreasing, which implies that they have inverse functions. These activities introduce students to theinverse operation of an exponential function (without naming it), and have students estimate values of thisinverse function and investigate the shape of its graph. The logarithm function is named and providesexamples of calculations with this function that do not require using a calculator. The change of baseformula is also included, which is motivated partially by looking at the ratio of logarithms for the samenumber using different bases. Students also explore the graphs of logarithmic functions. Because alllogarithmic functions are proportional to each other, students can observe common trends and utilize theirknowledge of scaling and translation of graphs. Two of the three laws of logarithms are introduced byexploring the relationship between the length of a password and its perceived strength. Students use thelog of a product rule to confirm patterns they see in changes to the number of characters used in passwordcreation, and conjecture about the log of a quotient. The logarithmic relationship between the Richterscale and the energy released by a seismic event is studied.

In these activities, students develop skills for interpreting the slope and vertical intercept ofa linear function by comparing costs associated with data usage from two competingwireless companies. They also develop the skills needed to reverse the process of a function inthe context of converting between different temperature scales. Students explore velocity/time graphs oftwo moving vehicles, estimate the time when the vehicles are traveling at the same velocity, and refinesolutions by algebraically solving for the intersection point of the two lines. Also, students investigate twoposition functions using graphing technology. They learn how to plot multiple functions on the same setof axes, how to set viewing windows appropriately, and how to use a graphing device to trace andevaluate values of functions. 

In these activities, students work exclusively with linear relationships. Students determine the rate ofchange between two variables and look for examples of a constant rate of change. They determine that therate of change is constant by observation or by calculating the change in the rate of change. Students userate of change information to write a linear function. Students also identify linear relationships by lookingat first differences and observing that they are constant. They see this by observation (looking at a columnof values) and by calculating second differences and seeing that those are zero. Students also composetwo linear functions and observe that the result is a linear function. Students will use information about alinear relationship to derive an equation of a line. The activity begins with an example that asks studentsto use slope and vertical intercept to find the equation of a line; the slope-intercept form is then formallydefined. Students then consider examples in which the slope is known but the vertical intercept is not.Students use information about a point on a line and the slope of that line to find the linear equation. Theoverall objective is to develop and use the point-slope formula, 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1). Students take differentapproaches to finding the equation of a line.

In these activities, students model the area of a rectangular garden plot as a function of the length andwidth, which are linear functions of 𝑥; the resulting area function is quadratic. Students practicemultiplying linear factors to obtain a quadratic equation in standard form, and they investigate someproperties of quadratic functions. Students will also explore various properties of quadratic functions inthe context of a model for projectile motion. They will see how the first differences of a quadraticfunction correspond to the average rate of change and, in this context, to the average vertical speed.Students also see how second differences can be used to determine whether a quadratic function canappropriately model a data set. Students explore various properties of quadratic functions in the context ofa model for projectile motion. They graph quadratic functions using technology and identify the vertex ofthe graphed function. Students also determine whether the vertex is a maximum or minimum value of thefunction and learn how to determine, without graphing, which functions have a maximum and whichfunctions have a minimum. Finally, students investigate a model for unit cost when the cost function is acubic polynomial. The resulting unit cost function is quadratic; students confirm this fact numerically,graphically, and algebraically. 

In these activities, students are introduced to power functions that have an integer exponent byconsidering a single application involving both second- and third-order power functions. Students are alsointroduced to integer-order power functions with negative integer exponents through the contexts ofstructural and civil engineering, as well as the illuminance of an object. Students extend the rules ofexponents to expressions containing fractional powers. Fractional powers are defined, and studentspractice simplifying expressions to see that the same rules of exponents hold. They continue to developskills in moving back and forth between radical notation and exponent notation. 

After a preliminary examination of the short-term behavior of rational functions, these activities turn tothe study of how rational functions behave for very large positive values of the independent variable.Students will build on these ideas by focusing on rational functions with horizontal asymptotes, that is,rational functions in which the degree of the denominator is greater than or equal to the degree of thenumerator. They interpret this behavior graphically as horizontal asymptotes. Students also explorerational functions whose graphs do not have horizontal asymptotes. The notion of a slant asymptote isintroduced.

In these activities, students use a cost/revenue context to interpret the meaning of the pointof intersection between two lines (i.e., break-even point) and begin estimating solutions ofsystems of two linear equations by analyzing their graphs. The activities also help students understandthat the number of solutions to a system of equations depends on the slopes and vertical intercepts of theequations in the system. By exploring three different scenarios, students realize that a system of equationsmay have one, infinite, or no solutions. Students learn how to use substitution to solve systems of linearequations algebraically and develop strategies that help them adjust the substitution method to solve anysystem of linear equations. Students will also learn about a second algebraic technique to solve linearsystems of equations, known as elimination by addition. Students apply their understanding of solutionsof systems of equations to the application of maximum heart rate. 

In these activities, students model the height of a projectile using a quadratic function. The formula forthis function is derived using scaling and shifting, starting with the function 𝑓(𝑡) = 𝑡2. They alsoinvestigate different algebraic representations for the same function by composing sequences oftransformations and inverting these sequences. Students practice using standard form, vertex form, androot form for the formula of a quadratic function in the context of a model. The activities reinforcestudents’ algebraic skills in solving equations and using the quadratic formula, and introduces them to theskill of factoring quadratic polynomials. The concept of total change as the area under a rate of changecurve is explored. Since the total change is quadratic when the rate of change varies as a (non-constant)linear function, many of the techniques of previous lessons for interpreting quadratic functions arereviewed.

In these activities, students solve exponential equations using the compound interest formula. Theyexplore the basics of savings bonds to see that the interest rate has the greatest influence on theinvestment. Students also apply logarithmic rules learned in previous activities to solve for the variable inan exponential equation. Students are introduced to the ideas of guaranteed savings bonds and how theactual value differs from the federally guaranteed amount. They are then introduced to the “Rule of 72” toapproximate doubling times of exponential functions quickly. Students also develop their skills at solvingequations that contain a single logarithmic expression or two or more logarithmic expressions and checkfor extraneous solutions.

In these activities, students extend their knowledge of function composition to rational functions. Studentssee that the domain of a composite function 𝑔(𝑓(𝑥)) is a subset of the domain of 𝑓(𝑥) and may be furtherrestricted by the domain of 𝑔(𝑥). Similarly, they see how asymptotes of the composite function 𝑔(𝑓(𝑥))are related to asymptotes of the functions 𝑔(x) and 𝑓(x). Students also continue to explore how change istransmitted through function composition. By exploring a distance-rate-time scenario, students realize theimportance of being able to add two rational expressions. Direct instruction leads students through thealgebra of finding a common denominator and helps them compute the final sum. Adding rationalexpressions that require a first step of finding a common denominator is focused on, as well. The need toperform the addition is emphasized as it allows one to solve certain contextual problems. Studentscontinue adding and subtracting rational functions to create new rational functions. Students focus onadding or subtracting rational functions in which the denominators share common factors. 

In these activities, students use tables and graphs to examine behavior near discontinuities of a rationalfunction. They also explore the end behavior of rational functions and relate the end behavior to STEMproblems.

In these activities, students work with functions that feature fractional (or rational) exponents. Theycontinue using the rules of exponents to simplify expressions involving fractional exponents. Studentsalso work with functions that feature fractional exponents, determine the inverse of such a function, andcontinue using the rules of exponents to simplify complicated expressions that involve exponents. Theyalso learn to identify whether a function has an inverse. Students study functions with fractionalexponents and their graphs. They explore the shapes of graphs of power functions 𝑦 = 𝑥𝑝/𝑞, where 𝑝 and 𝑞are integers, and describe properties of the graph depending on whether 𝑝 and 𝑞 are even or odd. 

In these activities, students begin learning the terminology of dependent and independent variables anddifferentiating between these two types of variables given algebraic or graphical representations offunctions. Students also start to connect the independent variable to input values of functions and thedependent variable to output values. Students discover that a function is more than a formula, table, orgraph, and learn that a function is a process that takes input values and assigns output values. Givendefinitions for the domain and range of a function, students also explore how arithmetic as well as contextcan affect domain and range. Finally, students will begin evaluating, graphing, and interpretingexpressions such as f(x + 1) and f(x) + 1.

In these activities, students are introduced to polynomial functions in the context of modeling the heightof an object and the motion of an object along a straight line. Students investigate these models usinggraphing calculators or apps and reflect on the advantages and limitations of modeling with polynomialfunctions. A method for finding a formula for a polynomial given its graph is also presented, and studentswill practice with multiplying polynomials. Students are also given additional exposure to modeling withpolynomials, investigating their graphs, and manipulating them algebraically. The problem of factoringpolynomials as an example of reverse engineering is also presented.

This resource contains a news article and a facilitation guide. Students practice data analysis and review the point-slope form of a line. This activity aligns with MATH 1342 Learning Outcome 2: Recognize, examine and interpret the basic principles of describing and presenting data

Upon successful completion of this assignment, students will
- formulate a researchable, open-ended question on a current and controversial topic that has two clear sides.
- construct a well-structured research proposal that includes credible sources and follows APA format guidelines, including a cover page, in-text citations, and a references page.

Author: Kimberly Stelly
Editor: Mary Landry, C. Anneke Snyder
Supervisor: Terri Pantuso

Upon successful completion of this lesson, students will
- compose a persuasive letter following the Rogerian argument form, including a summary of opposing views, a statement of position, a proposed compromise, and a conclusion.

Author: Kimberly Stelly
Editor: Mary Landry, C. Anneke Snyder
Supervisor: Terri Pantuso