Thank you for choosing the Dana Center Math Pathways (DCMP) Curriculum resource. The DCMP course programs are research-based and developed from the DCMP Curriculum Design Standards. To obtain the complete course, which includes instructional resources, rubrics, PowerPoints, and answer keys for the preview and practice assignments, you can visit the Dana Center Curriculum Resource Portal to request access. For a low-cost digital version that integrates seamlessly with most Learning Management Systems (LMS), you will need to fill out a Lumen Learning Online Homework Manager (OHM) request form. For any other questions, concerns, or support, please contact Charles A Dana Center danacenter@austin.utexas.edu. 

In these activities, students discover how the constant 𝑒 (Euler’s number) arises naturally whenapproximating an exponential function with a linear function. They use technology to confirm that at thepoint (0,1), the function 𝑓(𝑥) = 𝑥 + 1 is tangent to, and therefore a linear approximation of, 𝑔(𝑥) = ex.After confirming that, at 𝑥 = 0, the function 𝐿(𝑥) = 𝑥 + 1 is a linear approximation of 𝑓(𝑥) = 𝑒!, studentsuse this information to find more general approximations for exponential functions by scaling the input.These functions are applied to solving problems involving exponential growth and decay. Students alsodiscover how the constant 𝑒 arises naturally when modeling compound interest with increasingly frequentintervals of compounding. 

In these activities, function notation is introduced as a way to efficiently describe quantities and how theychange. Further, the importance of spending class time building a learning community and sharedresponsibility with each new group of students is encouraged. Students also practice using the functionnotation and immerse themselves in situations that have many changing quantities. Students will begin todevelop their own understanding of what a function is, what it is not, and how to identify functionrelationships. They will also continue to refine their strategies for recognizing function relationships andthink about the concept of a function in multiple representations. 

In these activities, students investigate rational functions near vertical asymptotes. The context is a modelfor the apparent height of a distant object. Models of the time to charge or discharge a cell phone batteryas a function of the charger current or screen size is used to investigate the output values of the functionnear its asymptotes. In the context of a model for the position of a shadow of a projectile, studentsencounter functions with removable discontinuities (corresponding to “holes”) and non-removablediscontinuities (corresponding to vertical asymptotes). Students develop strategies for determining thebehavior of rational functions near vertical asymptotes using algebra. 

In these activities, students understand that it is not always appropriate to use technology to graph rationalfunctions. They explore some of the problems that often arise when using technology to graph rationalfunction, and begin to understand why a hand-drawn graph is, at times, more appropriate than a computer-generated graph. These technology problems motivate the need for creating hand-drawn graphs and setthe stage for subsequent activities. Students also explore how to use technology to analyze rationalfunctions and to find extreme values. They set appropriate windows on their graphing technology to viewdifferent intervals of the graph to find local maxima and minima. Students create hand-drawn graphs withnon-constant scales that might better show all the features of a rational function, including those thatmight be difficult to see on a computer-generated graph. Students are introduced to an example of howrational functions are used in pharmacokinetics, a branch of pharmacology that studies substancesadministered to humans. Also, the equation for relativistic velocity is presented as supplemental contentand students use it to calculate observed velocities of objects traveling at speeds over 25% of the speed oflight.

In these activities, students begin to explore the inverse of a quadratic function. The inverse is thought ofas a function that “reverses” the process of the given quadratic function. Students refine theirunderstanding of inverse functions. They work with the formal definition of the inverse of a function, findinverses of linear functions, and encounter functions that do not have inverses. Also, students learn how todetermine a domain restriction in order to make a quadratic function one-to-one. They also learn to findthe inverse of the restricted function, such as finding the inverse of a square root function and making anappropriate domain restriction on the resulting quadratic function. Students are also introduced to theconcept of instantaneous rate of change, which they approximate by calculating average rates of changeover shorter and shorter time intervals. 

In these activities, students explore linear growth functions using information about different wildlifepopulations. Students learn key information about linear functions, such as slope, vertical intercept, andwhether the function is increasing or decreasing, using multiple representations. They also continue topractice using and interpreting function notation. Students will use multiple representations to interpretexponential functions as modeling growth or decline by an equal percentage over equal time periods.They also calculate the average rate of change of a function over a given interval. Students determine theperiod of a periodic function and will use the period to make predictions of future values of the function.Students continue to develop skills involving function notation, increasing/decreasing functions, andaverage rates of change. Students also determine whether given functions appear to be linear, exponential,or periodic, and then they make predictions based on their decisions, choosing among various strategies.Finally, students are introduced to study groups. In an active, collaborative learning environment, studentsare responsible for their own learning and for supporting the learning of others.

In these activities, students are introduced to piecewise-defined functions by having exploring the conceptof braking force and its relationship to the distance that a brake pedal is depressed. Within this context,students begin writing the equation of a piecewise-defined function using descriptions of its behavior overdifferent intervals. The activities also introduce students to an example of a logarithmic function. Studentsexplore this function by examining rates of change. Students use multiple representations of a logarithmicfunction (i.e., tables and graphs) in their investigations. The activities also continue to explore thelogarithmic relationship between the magnitude of an earthquake and the energy it releases. Studentscontinue to use multiple representations of a logarithmic function to investigate changes betweenindependent and dependent variables. They also show students that some functions may have less regularbehavior than other functions they’ve explored so far. All of the vocabulary and tools previouslyintroduced will still allow students to explore these functions, identify intervals where function valuesincrease and decrease, and calculate the average rate of change. 

In these activities, students investigate the logistic model for population growth. Unlike exponentialgrowth, the logistic model “levels off” at a carrying capacity. Therefore, the logistic model assumes thatthe surrounding environment can only support a certain population due to contextual constraints. Theequation associated with the logistic model contains exponential terms, similar to the models associatedwith Newton’s law of cooling/heating. Students will adjust a parameter of the model so it best matchesthe given data. The final model provides insight into the long-term behavior of the population. Studentsalso study logistic models, which lead to exponential equations that are algebraically more demanding tosolve than those in previous activities. 

In these activities, students develop and use a strategy to match an exponential function to existing data.Students are asked to determine an appropriate base, decay rate, and initial value (or scaling factor) of theexponential function and decide if the resulting model is reasonable. They continue to explore exponentialfunctions with a scenario involving an analog produce scale, which is typically encountered insupermarkets and outdoor farmer’s markets. Students will work with an unspecifiedfunction, which is oscillatory. They also examine the energy delivered by a defibrillator, which is acomposition of an exponential function with a power function. Composing these two functions yieldsanother exponential function that models the energy delivered by the defibrillator as a function of time. 

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.