AFM Curriculum Index

  • Linear Inequalities with a Parallelogram (NCSSM)
    Students are given the coordinate of a point in the second quadrant. Based on this point they develop a group of four linear inequalities whose solution forms a parallelogram that falls in the second quadrant of the coordinate system. The task asks students to find equations of lines and look at points of intersection. They learn to graph inequalities on the graphing calculator.
  • Linear Programming (NCSSM)
    Using a problem setting of varying hours of summer work with two possible jobs, students explore the restrictions of domain and range and linear inequalities to set boundaries on a region of the graph where solutions may lie. Within this region, they must determine the solution that gives the maximum weekly income.
  • Applications of Rational Functions (NCSSM)
    By developing a function to describe the annual cost of a refrigerator and given a function describing concentration of drug in the body, students relate the behavior of the graph of a rational function with the phenomenon it describes. Asymptotes and particular points become important information about the application.
  • Reading Graphs Links to an external site.
    This lesson presents students with situations that describe components of a graph-giving ordered pairs in terms of the real life setting. These activities can help weaker students connect written descriptions with mathematical information.
  • Caffeine in Your Body Links to an external site.
    This lesson introduces the use of the exponential function as a means to describe the elimination of caffeine in the body.

 

Advanced Functions and Modeling Standard Course of Study

Technology Connections Useful Throughout the Course:

  1. Data Flyer Links to an external site.
    The Java applet allows students to plot data and superimpose a graph over the data. There are no curve fitting capabilities, but can show deviations of a superimposed model.
  2. Function Flyer Links to an external site.
    This Java applet allows the student to enter a function, see its graph. All constants of the function become associated with sliders that allow students to investigate the effect of changing the values of the constants.

 

GOAL 1:  The learner will analyze data and apply probability concepts to solve problems.

1.01  Create and use calculator-generated models of linear, polynomial, exponential, trigonometric, power, and logarithmic functions of bivariate data to solve problems.

  1. Interpret the constants, coefficients, and bases in the context of the data.
  2. Check models for goodness-of-fit; use the most appropriate model to draw conclusions and make predictions.

Lesson Connections:

  • Guess the Age (NCSSM)
    Students are asked to guess the ages of a group of famous people. The actual age is paired with the student’s guess and a scatter plot of the ordered pairs (actual age, guessed age) is sketched. This is used to begin a discussion of information given in a scatter plot, information provided by a linear model (y=x), and the accuracy of using the model for prediction. In an attempt to identify the best guesser in the class, techniques are used that anticipate residuals.
  • Introduction to Linear Data Analysis (NCSSM)
    Using a collection of eight scatter plots, students determine graphs of lines or curves that might be appropriate models to describe bivariate data. Data that associates the length of a spring with the weight hung from the spring is used to find the linear regression model. Interpret the slope and the y-intercept of the line, and forecast other ordered pairs from the linear model.
  • Price of Apples (NCSSM)
    More practice with scatter plots, linear regression model, interpreting slope and y-intercept, and prediction. Students will shift data left to give meaning to the y-intercept. Students discuss possible criteria of linear regression model.
  • Hurricane Fran (NCSSM)
    The criteria of the linear regression model are defined. The Geometer's Sketchpad geometric illustration is useful to showing the sum of the squares of the residuals. The Fran data is definitely not linear. Fitting this data with a line will show the usefulness of the residual plot.
  • Football and Braking Distance: Model Data with Quadratic Functions (NCSSM)
    Students are given data to describe the trajectory of a football tossed from the tallest bleachers of a stadium. The data is fit with a quadratic function using least squares criteria. Given data extracted from page 288 of Glencoe's Algebra II book, students investigate braking distance versus speed of a car. Using quadratic least squares, the student finds a best-fit function for the data. Data is given on reaction distance versus speed of the car. When reaction distance is added to braking distance to find total stopping distance, students fit another quadratic function. A Follow-Up Problem relates number of sides of a polygon with the number of vertices to create a quadratic function.
  • Collecting and Fitting Quadratic Data with the CBL (NCSSM)
    Using the CBL and the graphing calculator, students work in groups to collect data describing the freefall of an object over time. The data collected includes data not relevant and that must be eliminated, and data is shifted near the y-axis to make the intercept meaningful. The students describe the meaning of the coefficients. The experiment is run again with an object that has drag (like a hat) and a model is found. The follow-up problem works with the football data from the lesson: Football and Braking Distance: Model Data with Quadratic Functions.
  • Polynomials as Models (NCSSM)
    A data set of the average price of gasoline for each year from 1993 to 2001 shows data with many changes. Using all the different regression curves and the regression line from the calculator, the students investigate the best model of the data and discuss its ability to predict.
  • Half-life and Doubling Time (NCSSM)
    Skittles or M&M's are randomly thrown onto a paper plate. The candies that fall with a letter face up are removed. Students document throw number and number of pieces remaining. Using the exponential regression fit, we find a decreasing exponential function with a half-life of one. A similar data collection activity that leads to an increasing exponential function with a doubling time of one results from cutting a sheet of paper, stacking the resulting pieces and cutting again. As a follow-up question, the Hurricane Fran data from the Algebra 2 indicators is fit with an exponential function. Given the students have half-life and doubling time problems in the lesson, they can then determine if this data has a half-life or doubling time.
  • Purchasing a New Car (NCSSM)
    Students visit the Kelly Blue Book web site to select the used car of their dreams. From the web site they document the value of this car for different model years. Using this data, they create a scatter plot, find a curve of best fit, interpret the meaning of constants, and make predictions.
  • The Box Problem (NCSSM)
    Students build open top rectangular boxes from a standard sheet of paper by cutting congruent squares from each corner. Data is collected that pairs the length of the side of the cut out square with the volume of the resulting box. To describe a clear pattern shown in the scatter plot, students develop a function through analysis of the box design. Based on this function, the length of the side of the square is determined that will create a box of maximum volume, and two squares that will produce a box of equal volume.
  • Hurricane Fran AFM (NCSSM)
    The exponential function models the repair of electricity outages for customers after Hurricane Fran. The model is found using data analysis and logarithms help students solve for additional information based on the model. The graphing calculator is used to find the model.
  • Voltage Problem (NCSSM)
    Using the Calculator Based Laboratory (CBL), the voltage from a capacitor is measured over the time since the capacitor was separated from the battery. Using data analysis and the exponential function students create a model of the decay of the voltage in the capacitor. The CBL and graphing calculator are used to collect the data and the graphing calculator is used to find the model. Sample data is included if no CBL is available.
  • Graphing Exponential Functions and e (NCSSM)
    A two-day lesson takes students through graphing exponential functions using the ideas of transformations. In the second part of the lesson the number e is developed using the idea of compounded interest and using the limit definition of e. A total of five days is allocated for this lesson. The graphing calculator is used as a learning tool throughout the lesson.

 

1.02  Summarize and analyze univariate data to solve problems.

  1. Apply and compare methods of data collection.
  2. Apply statistical principles and methods in sample surveys.
  3. Determine measures of central tendency and spread.
  4. Recognize, define, and use the normal distribution curve.
  5. Interpret graphical displays of univariate data.
  6. Compare distributions of univariate data.

 

Lesson Connections:

 

Technology Connections

 

1.03  Use theoretical and experimental probability to model and solve problems.

  • Use addition and multiplication principles.
  • Calculate and apply permutations and combinations.
  • Create and use simulations for probability models.
  • Find expected values and determine fairness.
  • Identify and use discrete random variables to solve problems.
  • Apply the Binomial Theorem.

Lesson Connections:

 

GOAL 2:  The learner will use functions to solve problems.

2.01  Use logarithmic (common, natural) functions to model and solve problems; justify results.

  1. Solve using tables, graphs, and algebraic properties.
  2. Interpret the constants, coefficients, and bases in the context of the problem.
    • Radio Dial (NCSSM)
      Students create a data set of the distance between the end of the radio dial and the location of a specific frequency on the dial. Using their knowledge of functions and a graphing calculator the students create a model for the geometry of the dial.


2.02  Use piecewise-defined functions to model and solve problems; justify results.

  1. Solve using tables, graphs, and algebraic properties.
  2. Interpret the constants, coefficients, and bases in the context of the problem.

Lesson Plans:

  • Piecewise Defined Functions as Models (NCSSM)
    The data for the Olympic swimming records for the 400 Freestyle found in the Algebra 2 Indicators show data with two definite trends over time. In this lesson students develop a piecewise-defined linear function using domain restrictions and the linear regression model. This model provides specific information in the slopes to compare the data of the two trends. Since both men's and women's data is given, one data set can be discussed in class and students can follow up the lesson with the other data set.
  • New Jersey Turnpike (NCSSM)
    Mileage traveled and charges for travel along the New Jersey Turnpike produce a piecewise defined linear functions. There is discussion of whether outliers should remain in the model, and if new exits were added what charge would be appropriate. The graphing calculator is used to find the models for the data.

 

2.03  Use power functions to model and solve problems; justify results.

  1. Solve using tables, graphs, and algebraic properties.
  2. Interpret the constants, coefficients, and bases in the context of the problem.

Lesson Plans:

  • Equations with Radical Expressions (NCSSM)
    Data representing the period of a swinging pendulum versus the length of the pendulum can be best modeled by a square root function. Data and an appropriate model are both given to the students. Questions from the NC Algebra II Indicators require students to solve equations involving radical expressions. Solutions are also investigated from both a graphical and an analytical point of view.

 

2.04  Use trigonometric (sine, cosine) functions to model and solve problems; justify results.

  • Solve using tables, graphs, and algebraic properties.
  • Create and identify transformations with respect to period, amplitude, and vertical and horizontal shifts.
  • Develop and use the law of sines and the law of cosines.

Lesson Plans:

 

2.05  Use recursively-defined functions to model and solve problems.

  1. Find the sum of a finite sequence.
  2. Find the sum of an infinite sequence.
  3. Determine if a given series converges or diverges.
  4. Translate between recursive and explicit representations.

Lesson Connections:

  • An Introduction to Arithmetic and Geometric Sequences Links to an external site.
    This lesson encourages students to build sequences based on the parameters of a beginning value, a constant multiplier and a constant addition. The result of setting these parameters is a plot of (sequence #, value) and a list of values. There is a handout that students complete on the effect of each parameter. This notion of recursion or sequencing is connected to fractals and Pascal's triangle in follow-up lessons. This lesson might be used for introduction to sequences. Concerns: there is little attention paid to analytical statements of the sequence. Notation is not developed at all. On the plot of the results of a sequence the points are connected.
  • Pass the Candy (NCSSM)
    Each student sitting in a circle gets some candy--different amounts for each. Students are asked to give one-half of their candy to the person on their right. After several iterations, a pattern emerges. After running the experiment, students model the situation using recursively defined functions. Sequence mode and matrices on the graphing calculator are used to describe the phenomena.
  • The Great Lakes Problem (NCSSM)
    Recursively defined functions model the flow of pollution into Lakes Huron, Erie, and Ontario. By studying how pollutants flow form one lake to next and issues of equilibrium, students investigate how to decrease pollution. The sequence mode of the graphing calculator is used approximate the functions.
  • Using Algebra and Discrete Mathematics to Investigate Population Changes in a Trout Pond
    Using recursion the population of a trout pond is modeled. Aided by algebra and hints if needed, students transform the recursively defined functions into explicit functions to investigate long term effects.


Optional functions for additional or continued study: linear, quadratic, cubic, exponential, rational, parametric, and linear programming.

  • Calculate When the Oil Industry Will Dry Up Links to an external site.
    Connections to resources allow students to use models to determine when a country's oil will dry up. Further questions are asked about how the US will react to this dilemma.

  • Ring Toss (NCSSM)
    A ring toss game is used to introduce parametric equations. Students model the toss by describing the horizontal and vertical components of the throw. The graphing calculator is used for graphs and particular values.
  • Parametric Equations (NCSSM)
    A ring toss game is used to introduce parametric equations. Students model the toss by describing the horizontal and vertical components of the throw. The graphing calculator is used for graphs and particular values. The second problem—a more advanced problem—describes the motion of the Land Rover on Mars.


Websites with possible modeling problems: