Problem Based Learning

Jackson Huckaby

"The human overpopulation issue is the topic I see as the most vital to solve if our children and grandchildren are to have a good quality of life."

Alexandra Paul

**Brief
Description:** Students will
collect population data and apply an appropriate mathematic model to inspire
discussion about the future possibilities of growth of a region. Students will
then discuss needs that should be addressed and make proposals for the
government to consider.

**The Essential Question to
Be Answered:** **How can we collect data from the past and
use mathematics to make inferences and decisions that can better our future?**

**Grade Level: 10 ^{th}**

**Subject Area: Mathematics
II**

**The Authors: Jackson
Huckaby**

**Problem-Based
Performance Guide**

The situation
is... Today’s society is clearly
focused on the now. Communities are growing in all directions and are building
new towns and cities across the entire country.

The problem
is... While everyone is building buildings and cutting down trees, little
thought is being exercised about our future and our natural resources. The
government you live in is meeting a month from now to discuss growth.

Your task is
to... Collect accurate data about the population growth of your region over the
last century (if applicable). Use this data to develop an accurate algebraic
model to demonstrate the future growth trends for the region. Organize your
information and prepare a presentation for the government meeting next month.

You will know
you have successfully finished when... You have fully developed your future
growth model for the region; you have come up with suggestions that should be
followed to overcome any particular obstacles that coincide with said growth
model; and you have presented your discoveries with the class.

Challenges you
may encounter are... societies lack of care about the future. Getting adults to
listen to the thoughts of teenagers. (This is why you are bringing the
mathematical evidence!!)

Resources you
can depend on are... TI 83 or similar graphing utilities, Excel,
http://www.census.gov/ , google

You will
need... graph paper, colored markers, poster board, tape.

The final completion
date is...

2 weeks from today.

Go!

The focus of the lesson will be using exponential functions to model population growth in a region. The students will be asked to make inferences based on their model and to discuss limitations of the model. Each group will be asked to pick a particular region that they wish to work with; this can be a county, a city, a state, or a country. They will then need to go online and access the population data from this region; this can be done with the US Census information. After collecting data students will represent this data both in a table and graphically. Students will use this information to discuss the overall trend of the population.

1.Students will first be asked to apply a linear fit to their model.

They will be asked to discuss its strengths and weaknesses.

2. Next students will use appropriate technology to fit an exponential model to the data.

Students will be expected to discuss domain and range in context of the situation and discuss limitations of the model.

3. Students will need to reflect on the following:

a. What variables of the region will sustain or affect the trend of the population?

b. How will this trend affect the future of the region and what needs will need to be addressed?

c. What are some potential problems that could develop because of this trend?

Students will be expected to:

- Appropriately fit an exponential model to the data.
- To accurately graph their data points and have to exponential function fitted on the graph.
- Students will also have to have written out this function at the bottom of the graph.
- The students must also have accurately provided context to the domain and range of the function.
- It should be obvious that the students understand what the model is saying and the limitations of the model. - This is where the EQ comes in to play, the purpose of the task is to use mathematics to model population growth and then be able to draw conclusions and make predictions about the future.

The Georgia Performance Standards

**MM2A2. Students will explore exponential functions.**

a. Extend properties of exponents to include all integer exponents.

b. Investigate and explain characteristics of exponential functions, including domain and range, asymptotes, zeros, intercepts, intervals of increase and decrease, rates of change, and end behavior.

c. Graph functions as transformations of *f(**x) *= *ax.
*

d. Solve simple exponential equations and inequalities
analytically, graphically,

and by using appropriate technology.

e. Understand and use
basic exponential functions as models of real phenomena.

**MM2P1. Students will solve problems (using appropriate technology).**

a. Build new mathematical knowledge through problem solving.

b. Solve problems that arise in mathematics and in other contexts.

c. Apply and adapt a variety of appropriate strategies to solve problems.

**MM2P3. Students will communicate mathematically.**

a. Organize and consolidate their mathematical
thinking through communication.

b. Communicate their mathematical thinking coherently
and clearly to peers, teachers, and others.