Comparing Exponential and Linear Functions Day 2

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Objective

SWBAT write functions that model exponential and linear growth and make comparisons between the two.

Big Idea

This lesson is designed to show that an exponential functions growth will eventually exceed the growth of a linear function.

Warm Up

10 minutes

This activity gets students thinking about the comparison between linear and exponential functions discussed previously.  Students will work in pairs on this matching activity.  Cut out the eight cards from exponential_v_linear2_matching and instruct students to sort them into two categories: linear and exponential.  

Once students have sorted their cards, I will choose one card at a time and have a pair of students justify why they put that card in the group they did.  Other students can listen to the response and either try to change their classmate's mind or add on to their reasoning with another idea. I will let 2-3 students speak for each of the 8 cards so that most or all members of the class get to participate. I'll use talk moves such as, "Johnny, can you add on to what Melissa just said?" Or "Johnny, can you explain what Melissa just said in your own words?"  

Investigation

25 minutes

This question is meant to help students see that increasing the starting point or growth rate (or both) for a linear function will not make exceed the growth of an exponential growth function in the long run.  Students will investigate this concept in this contextual application about food shortages (exponential_v_linear1_launch).  To add context, I have included a headline from 2012 in the resources.  This news item shows that food shortages really are a major concern in the world today.

Students will first work with a partner to read and understand the problem.  They will then try to determine how to represent the two scenarios (food supply and population growth) mathematically.  Students will initially struggle with the values in the millions.  Some students may try to write the food supply function using the number 4,000,000.  While this will still be correct, it is somewhat inefficient.  I plan to guide students to understand that since the food supply and population growth are both in millions the number "4" can represent the quantity 4 million (MP1, MP2).    

Students should have access to graphing software or a graphing calculator for this investigation. In this case we will be using technology to help students' answers to be as accurate as possible (MP5). I encourage students to include sketches of their graphs to show the relationship between the food supply and population growth.  Students can also use the intersect feature on their graphing calculator or software to determine when the food shortages will occur.

Once students have had time to investigate the three different scenarios, I will have several of the groups share their findings with the class.  In order to get more students involved, consider having groups share out only one portion of the question (example only part a).  This way, as different groups come up to present the class will continue to see that a shortage will eventually occur.   

 

Closure

5 minutes

This Close has two different prompts.  Student's reasoning about the first will let you know if they are grasping the major theme of this particular lesson.  Some students may reason about this question quantitatively by looking at future values beyond those that they investigated in class.  Others may look at the two graphs and how they could be manipulated further. Encourage students to include a thorough explanation of their thinking.  This will help you gather important insight into their understanding.

The second question brings the scenario back to reality.  The answer to this question is totally opinion based, but could be grounded in the day's investigation.  I would look for students to think logically about this question.  Ideas like:

  • Does the population of an area really grow at exactly the same rate over 100 years?
  • Can increases in food supply be promised for 100 years in advance?
  • What happens if there is a poor growing season?

The point of this question is for students to realize that mathematical models have great utility but the domain of the model needs to be a consideration.