Advanced Algebra (Mathematical Modeling)

In this course, the primary focus is the creation of linear and exponential models to represent rates of growth involving a variety of situations, particularly human population growth.

The course addresses the following focus questions:
*How do rates of change relate to the steepness of a graph?
*How do we use graphs to interpret data?
*What is the definition of slope and how does it relate to real-life situations?
*How do we represent slope graphically and algebraically?
*What is the relationship between slope and a derivative?
*What real-life situations illustrate exponential growth?
*How do we represent exponential functions in terms of different bases?
*How do we convert an exponential function to a base of e?
*How can we adjust an exponential function to fit it to a set of data?

In addition, the following standards are addressed and assessed throughout the unit:
*Evaluating average rates of change
*Understanding the relationship between the rate of change of a function and the appearance of its graph
*Using exponential functions to model real-life situations
*Developing an algebraic definition of slope
*Understanding the significance of a negative slope
*Seeing average speed as the slope of a secant line
*Developing the concept of the derivative of a function at a point
*Seeing that the derivative of a function at a point is the slope of the tangent line at that point
*Finding numerical estimates for the derivatives of functions at specific points
*Observing that the rate of change in population is proportional to the population
*Discovering any exponential function can be expressed another number as the base
*Learning that the value e is the same number as the special base for exponential functions
*Strengthening an understanding of logarithms
*Using an exponential function to fit a curve to numerical data
*Summarizing ideas about linear and exponential growth

Readings: 
"Small World, Isn't It?" - Interactive Mathematics Program
"Population: 7 Billion" - National Geographic Magazine
"Fatima's Story" - ZPG Population Education Program
Research: 

Students research population factors that affect the population growth in their selected countries from various sources considering both quantitative and qualitative data.

Media Used: 
"World in the Balance" - NOVA DVD
"Aftermath: Population Zero" - National Geographic
http://www.unicef.org/statistics/index_countrystats.html
http://www.xist.org/cntry/
http://www.prb.org/DataFinder.aspx
Interim Assessments: 

Students explore mathematical patterns in world population data and make forecasts to extend the patterns. Students organize their thinking in writing to present to the instructor and to their peers.

Students analyze graphical representations of linear and exponential sequences and make inferences based on the behavior of the graph. Additionally, students interpret statements about the graphs of different sequences, justifying their thinking in writing.

Students construct an addition (linear) and a multiplicative (exponential) model representing new computer sales. Students graph the data, calculate the discrepancies between their mathematical models and the actual data, and then interpret which model is better for forecasting future sales. Students then forecast sales into the future and explain some reasons why forecasting future sales would be important to the company.

Significant Assignments: 

Students learn and analyze how scaling can affect the visual representation of graphs through the creation of several graphs utilizing the same data set.

Students find “addition” and “multiplication” growth numbers to increase a given number to a specified target in a specific number of steps.

Utilizing a mathematical model, students utilize growth numbers to calculate intermediate values and then extend the model into the future or conversely into the past.

Students represent sequences algebraically working with the concepts of inverse operations, reciprocals, multiples, and powers.

Students graph addition (linear) and multiplicative (exponential) sequences and make comparisons including how the value of the growth number affects the shape of the graph.

Students construct secant lines and tangent lines on the graph of an exponential sequence and differentiate between them.

Significant Activities or Projects: 

Students research demographic data about their native countries and complete a population pyramid representing the age groups. Students interpret their population pyramids by making inferences about factors (social, religious, political, cultural) that could impact the shape of their pyramid.

Students construct and analyze growth spirals representative of addition (linear) and multiplicative (exponential) functions. Students display and present their work on chart paper, indicating visual characteristics of each type of spiral.

Students calculate the discrepancy between mathematical models representing the population growth of the United States and interpret which model fits actual data more closely, and then make resulting forecasts.

Sample PBATs: 
Native Country Population Project: Students tabulate the actual population data of their native country using the time period from 1950-2000. Students calculate the rate of growth under both a linear and exponential model and compare the resulting forecasts to assess which is closer to the actual population data both numerically and graphically. Students construct the secant lines and tangent line on their graphs and predict the rate of change at a specific year. Students then use the graphing calculator to find an exponential regression model to compare to their prediction. Students also use an exponential base of e, to forecast future population data and to predict when their country if continuing at its current rate of growth would be completely filled with people. Students also reflect on the social, political, and cultural factors that may affect their mathematical forecasts.