# We the People

## Linear vs. Exponential Growth

K20 Center, Kate Raymond | Published: November 18th, 2022 by K20 Center

• Subject Mathematics
• Course Algebra 1
• Time Frame 2-3 class period(s)
• Duration 135 minutes

### Summary

Students model population growth across continents to discover the difference between linear and exponential models. Students should already be familiar with both linear and exponential functions and equations.

### Essential Question(s)

How can population growth be modeled mathematically?

### Snapshot

Engage

Students read a brief news article about Boley, Oklahoma, and engage in a discussion about the relationship between the population of the town and how people live and work there.

Explore

Students gather data about the population of the United States. They then create both linear and exponential models for the data they collect and compare the two models for accuracy and precision.

Explain

Students determine whether linear or exponential models more accurately describe changes in population.

Extend

Students create exponential models for populations of five continents and make predictions based on these models.

Evaluate

Students read a brief article that introduces the concept of carrying capacity and create an exponential model of the world population to make a prediction about the number of years remaining before the Earth reaches its carrying capacity.

### Materials

• Internet access

• Microsoft Excel or graphing calculators

• Continent Populations handout (attached)

• Comparing Continents handout (attached)

• Carrying Capacity handout (attached)

• We the People Data Sheet (attached)

• Sticky notes

• Highlighters

• Pencils/pens

• Four Corners signs (attached)

• Markers (Mr. Sketch, Sharpie, etc.)

### Engage

Direct students to the article about Boley, Oklahoma, or pass out printed copies of the article. Have students complete a Why-Lighting activity to answer the question "Why did the author of this article include information about the population of Boley? What point do you think they were trying to make?"

Have students complete a Think-Pair-Share activity over what they highlighted in the reading. Ask students what consequences the declining population of the town has on the remaining residents. Discuss this with them to illustrate the relationship between population and the everyday lives of citizens.

### Explore

Distribute a copy of We the People Datasheet for every student. Have students work in groups of two or three to gather the data needed to complete the population column of the chart for Part A.

Have students create a line of best fit for the data.

Students should now complete Part B on the document by using the line of best fit they created in Excel or on their graphing calculators. They can enter the equation for the line of best fit given on their chart in the appropriate space indicated; then, they should use the formula to enter the values predicted by this model into the spaces of the chart.

Before students complete Part C, discuss the concept of error with your students. Ask students to create (or recall) a formula to measure error. Discuss this formula until the class agrees on an accurate formula. Students should record this formula on their worksheets and then complete Part C.

After students complete Part C, ask the class if they believe their linear model was accurate. Have them justify their reasoning orally.

Ask students to describe the physical meaning of the slope in the line of best fit. Lead them in a discussion until they reach the conclusion that the slope represents the change in the number of people living in the US each year.

Ask students to consider the population of the US in 1900 and 2010. Based on these populations, is it reasonable to expect that the increase in the number of people in the US was the same (approximately 2 million) each of these years?

Ask students if they believe a linear model accurately represents population growth. Hopefully, they are convinced that it does not.

Ask students to think of types of functions that might better fit population data. Lead them in a discussion until they propose exponential functions as a possible alternative. Have students give reasons exponential functions might provide a better fit, based on their prior knowledge.

Demonstrate how to create exponential curves of best fit on the TI-84 and in Excel. Directions on how to make exponential curves of best fit on TI-84 and on excel are attached to this lesson.

Have students complete parts D and E of the worksheet in groups of two or three.

### Explain

After all groups have completed their worksheets, have students discuss whether they believe the exponential model better fits their data in their groups. Then, have each group share the equation of the exponential model they created and the conclusions they reached. Record the equations of the models each group created on the board.

After each group has shared their model, write the generic equation for an exponential model (P=C(e^(rt)) or P=C(b^t)) on the board. Ask students to identify the value of C in their model and then to explain the significance of C.

If using TI-83 or TI-84, skip to the next step. If using Excel, have each group calculate the value of e^r for their specific value of r. Students should label this value b for base.

Have each group report out their value of b. Ask the class what they believe these values of b represent. Lead the class in a discussion until they reach the conclusion that the value of b represents the percent change in population per year and is written as one plus the decimal equivalent of the percent change.

Ask students to once again consider the years 1900 and 2010 and determine if it is reasonable to believe that the percent growth rates are equivalent for these two years.

Ask students if they believe exponential functions can accurately model population growth. Have them explain why they do or do not think the exponential models are accurate.

### Extend

Hand each student two sticky notes. Write the names of the five major continents (Africa, Asia, Europe, North America and South America) in a horizontal line across a whiteboard. Direct students to place one sticky note above the name of the continent they believe has the largest population and the other sticky note below the name of the continent they believe has the lowest population.

Write the following prompt on the board: "An exponential model is written P=Ce^(rt). Both C and r represent unknown values. Since there are only two unknown values for this model, only two data points are needed to write an accurate exponential model for a given situation." Use this prompt to complete a Four Corners activity. The signs needed for the Four Corners activities are attached to this lesson.

Tell students you are going to try to write models given only two data points and determine if the models created are reasonable. Group students into pairs or threes and give each group one section of the attached handout "We the People - Continental Populations." Each section contains data about a different continent, so be sure to distribute the sections evenly so that there are approximately the same number of groups working on each continent's data.

Have students work together to complete their section of the worksheet. Then, have all groups who are working on the same continent meet to share their results. Each content group should discuss their data and analysis until they agree on all solutions. This entails choosing a single equation to model the population of the continent.

Pass out the We the People - Comparing Continents handout if it was not copied onto the back of the We the People - Continental Populations handout.

Ask a group to share the model they decided upon and their reasoning. Ask each group whether they think the model they created is an accurate model for the population growth of their continent.

Have all students copy the model given for each continent onto the We the People - Comparing Continents handout. Students can then work with their groups to find the current population of that continent. Have all groups record their answers using clickers or whiteboards as a formative assessment.

Continue calling on continent groups until each continent has been discussed.

Have students work in groups of two or three to answer questions 1-10 on the handout.

Once all students have completed their handouts, as for volunteers to share their answers. Allow students to check their work on questions 1-10, discussing the solutions when necessary.

Refer back to the results of the sticky notes predictions. Ask students to consider the predictions that the class made as they respond to question 11 on the handout. Give the students several minutes to respond to this question, then discuss their responses with them.

### Evaluate

Distribute the We the People - Carrying Capacity handout. You may choose to have students complete this assignment individually or in pairs. They can either complete it in class or at home over several nights.