How Clean is the Water in Your Town?

(1) The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. Following are performance descriptions.

(A) The student describes independent and dependent quantities in functional relationships.

(B) The student gathers and records data, or uses data sets, to determine functional (systematic) relationships between quantities.

(3) (B) Given situations, the student looks for patterns and represents generalizations algebraically.

§111.33. Algebra II

(f) Exponential and logarithmic functions: knowledge and skills and performance descriptions. The student formulates equations and inequalities based on exponential and logarithmic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. Following are performance descriptions.

(3) For given contexts, the student determines the reasonable domain and range values of exponential and logarithmic functions, as well as interprets and determines the reasonableness of solutions to exponential and logarithmic equations and inequalities.

(4) The student solves exponential and logarithmic equations and inequalities using graphs, tables, and algebraic methods.

(5) The student analyzes a situation modeled by an exponential function, formulates an equation or inequality, and solves the problem.

§111.35. Precalculus

(3) The student uses functions and their properties to model and solve real-life problems. The student is expected to:

(A) use functions such as logarithmic, exponential, trigonometric, polynomial, etc. to model real-life data;

I. Overview

The overall concept of “Exponential Functions” is for the students to get an introduction to exponential functions, to understand exponential functions, and how they relate to real world applications. The class first explores the world population since 1650. Students then conduct a simulation in which a population grows at a random yet predictable rate. Both situations are examples of exponential growth. I feel this concept is important to know because it demonstrates a real life application of mathematics in nature, and helps to explain why we can only sustain so much life in a certain area. We will see that nature is a delicate balance, and disruptions in that balance can be catastrophic. This will be extended to the students’ “How Clean is the Water in Your Town?” project because it will be related to the exponential functions and growth curves students will see in bacteria in water. Furthermore, the students will need to pick the water purification technique they feel is best, and demonstrate the efficiency of their technique based on the proportions and ratios of clean water the technique makes. Some research has shown that some of these techniques produce clean water in an exponential function.

II. Performance or learner outcomes

Students will be able to:

a.) Compare and contrast linear and exponential functions

b.) Use a simulation to explore growth

c.) Describe the graphs of exponential functions

d.) Relate exponential functions to their water project

II. Resources, materials and supplies needed

a.) Copies of Skeeter handout

b.) Graphing Calculator for each student (approximately 20 students)

c.) Skeeters (tokens or candies with a marking on one side)

d.) A large, flat box (one for each group of 4 students)

e.) Data images of population growth

IV. Supplementary materials, handouts. (Also address any safety issues

Concerning equipment used)

All materials are listed above. As far as safety goes, we will be a big group of high school students and 1 teacher in a classroom of 20 or so students. Safety rules are those set by the teacher/school for the classroom and should be followed. All students should treat each other with respect. The materials, i.e. the calculators and skeeters, should also be treated with care and respect.

Five-E Organization

Teacher Does Probing Questions Student Does

Engage:

Learning Experience(s)

Welcome students to class. Give the students an overview of what we are going to be doing/learning today, what they will be doing, and hopefully what they will discover. 1. Divide students into groups of four. In each group, assign the roles of captain, recorder, reporter, and timekeeper.2. Explain that the class will examine population growth.3. Distribute the handout.

4. Have students read aloud the introductory paragraphs for the lesson.5. Ask students to consider what things they can do mathematically to make predictions about the future. Students should suggest that collecting data, making graphs, and looking for patterns would be useful in making predictions.

6. Show data images attached regarding the world population from 1650 to 1850.

7. In groups of four, have students describe any patterns they notice in the changes in world population from 1650 to 1850. 8. Have student groups predict the world population in 1950. 9. Have reporters state their team's prediction for 1950. Record the various predictions on the chalkboard or overhead projector. Be sure that students explain how they made the prediction, and have students discuss the various predictions. (Once students agree on which predictions are reasonable, you may wish to have them take the average of these predictions to come up with a whole class prediction.)10. Have student teams plot their 1950 point on the graph containing the points for 1650, 1750, and 1850. Because the three points for 1650, 1750, and 1850 lie somewhat along a straight line, have students check the reasonableness of their prediction by noticing if it lies along the same line. 11. Reveal the actual population in 1950. (The student predictions will likely have been much lower.) Then, ask them to use this new information to predict the world population in 2000. Again, have students discuss this problem in their groups. 12. Record the groups' predictions for 2000 on the chalkboard or overhead projector. Be sure to have students state how they arrived at their predictions, and allow them to discuss the reasonableness of these predictions. 13. Reveal the actual population in 2000. 14. Explain that things can grow in different ways, following different patterns and in ways we might not expect. Consequently, we adjust our predictions based on new information. Explain that students will conduct an exploration.

Critical questions that will establish prior knowledge and create a need to know

Any questions before we begin?

What things can we do mathematically to make predictions about the future?

Can you all describe any patterns you notice in the changes in world population from 1650 to 1850?

What do you think will be the world population in 1950? Reporters tell me your team's prediction for 1950.

You can see that the three points for 1650, 1750, and 1850 lie somewhat along a straight line. Is your groups prediction reasonable based on this observation?

After revealing the actual population, ask students if they were shocked, right, confused…Their predictions will most likely be much lower.

In your groups, can you make a prediction for the world population in 2000?

After revealing the actual population, ask students if they were shocked, right, confused…

Things can grow in different ways, following different patterns and in ways we might not expect. Consequently, we adjust our predictions based on new information. Explain that students will conduct an exploration.

Expected Student

Responses/Misconceptions

Students should suggest that collecting data, making graphs, and looking for patterns would be useful in making predictions.

Students should see an increase in population, and then a leveling off, and they may or may not know as to why.

Be sure that students explain how they made the prediction, and have students discuss the various predictions. (Once students agree on which predictions are reasonable, you may wish to have them take the average of these predictions to come up with a whole class prediction.)

Students may be somewhat shocked and confused as to why their predictions are so much lower, or higher, or whatever. Having them discuss their reasonableness and how they come up with their predictions is important here.

Explore:

Learning Experience(s)

1. Have a student read the directions for the exploration:

To help make predictions in real-world situations, researchers often use experiments known as simulations. The results of the simulations are gathered and analyzed. This data is then compared with known information about the actual population. If the result seems questionable, the simulation may be revised. 2. Have students explain what a simulation is in their own words. Elicit from students that a simulation is a model of a real-world situation.

3. Have several students give examples of simulations. 4. Have a student continue reading under the Exploration section:

This modeling process can be summarized by the following five steps:

1 creating a model

2 translating the model into mathematics

3 using the mathematics

4 relating the results to the real-world situation

5 revising the model

In the following exploration, you investigate this modeling process using a population of Skeeters.

5. Have students read the directions for the exploration. Then, give them 30 minutes to run the simulation and complete the portion of the handout under the heading Discussion 1.

6. Record the teams' predictions for "Shake 20." (Students make this prediction in Discussion 1, Part c.1 of the handout.)

Critical questions that will allow you to decide whether students understand or are able to carry out the assigned task (formative)

Any questions before we begin our exploration?

After reading the definition of a simulation… What do you all think a simulation is? In your own words…

Can you give me any examples?

While students are participating in the exploration, I will walk around and answer any questions in the groups.

Expected Student

Responses/Misconceptions

Students will probably get to the idea that a simulation is a model of a real world situation, and that we can use simulations when its harder to observe the real world situation.

Examples can include activities the students have seen so far in their Clean Water project.

Explain:

Learning Experience(s)

The students will regroup, and present what they have found. 7. Lead a class discussion about the predictions. Have students explain the patterns they noticed as they ran the simulation, and how they used those patterns to make their prediction.

8. Give students 10 minutes to complete Discussion 2. (In Discussion 2, student teams decide the best way to describe the shape of the graph.)

9. Ask the reporter from each team to share the team's description of the shape of the graph. Record their descriptions on the chalkboard or overhead projector. Discuss the descriptions and elicit from students that the graph is a curve.

Critical questions that will allow you to help students clarify their understanding and introduce information related to concepts to be learned

Any last minute concerns, comments, questions?

What patterns did you notice? How did you use the patterns you observed to make your prediction?

What did you graph look like? What shape do you think it looks like? Why do you think it looks like that? What does that tell us about how much water we need?

Expected Student

Responses/Misconceptions

Students should see that the graph is a curve, and see the patterns in the results.

Students should discuss why it looks like a curve, mainly because population has to level off. It shows that in order to sustain life in a certain area, we need certain resources, and if there are too many people growing at a constant rate, these resources will eventually be depleted, so population tends to level off.

Extend / Elaborate:

Learning Experience(s)

Exponential functions are a good introduction to something called exponential growth curves. Ask the students to tell what they think are exponential growth curves, based on the previous activity. Ask how do they think exponential growth curves relates to their water project. Remind students that part of their water project is to pick the cleaning technique they think is best. Some research has shown that some of these techniques produce clean water exponentially from what is put in. Therefore, exponential functions come in handy to understand. Students should write down in their portfolios the criteria they will use when determining which technique they feel is “best” and why.

Critical questions that will allow you to decide whether students can extend conceptual connections in new situations

What do you all think are exponential growth curves, based on the previous activity? What does it sound like it’s about? How do you all think exponential growth curves relate to our water project?

What will you look for when deciding which cleaning technique is the best? Why do you choose those criteria?

Expected Student

Responses/Misconceptions

Based on the words, students should elicit that exponential growth curves are the curves we observed in our population growth, or at least think this. We will go over exactly what they are and why in a later lesson. Furthermore, they should relate exponential growth curves to the bacteria in water lesson they should have covered in their science class.

Students will most likely choose a technique based on how much clean water is needed in proportion to their towns population. We will discuss proportions later. Furthermore, they may wish to know how much water is cleaned in relation to how much dirty water is put in.

Evaluate:

Lesson Objective(s)

Learned (WRAP –UP at end) -> Summarize

Overall, we saw that population can grow exponentially, and how it levels off. Furthermore, we related this to our water project, and have a good starting off point to ratios, proportions, and exponential growth curves.

Can you summarize what you have learned? Any last minute questions, concerns, comments?

Give students the remainder of the class period to record what they learned in their project portfolios.

Critical questions that will allow you to decide whether students understood main lesson objectives

Any last minute questions or concerns or comments?

Thank you all for your attention and participation today.

Expected Student

Responses/Misconceptions

Students will summarize what they discovered and ask any last questions or concerns they have. Comments are welcomed, and students can write these down on the back of their worksheets before returning them to me.