Rates of Change, Exponential Growth, and Logarithms

Rates of change, exponential growth, and logarithms:

The effects of temperature on caterpillar growth rate

Subject: Algebra I/II ,Life Sciences Grade: 9-12

Topics: Exponential growth/logarithm functions, Ecology

Lesson plan designer: Melissa DeSiervo ()

Overview: This lesson and activity uses biological data on the response of organisms to temperature change to illustrate the mathematical concepts of rates of change, exponential growth, and logarithms. The first lesson includes a brief lecture (with accompanying PowerPoint) which introduces students to core biology and math concepts, and a worksheet where students answer questions and practice simple calculations. In the second part of the lesson, students learn about rapid environmental change in the Arctic and speculate how insects may respond to increasing temperature. Using a dataset on Arctic caterpillar growth rates, students will complete an activity in Microsoft Excel graphing exponential and logarithmic functions, and calculatingRelative growth rate (RGR) and Doubling time (DT).

Objectives: Students will learn to plot biological data, use simple mathematical models to describe the biological data, interpret exponential models, use Microsoft Excel, and understand the relevance of math to describe how Arctic systems are changing.

Estimated Time:The recommended time is two class periods. Alternatively, Lesson 1 can be completed as a stand-alone activity, and/or the Microsoft Excel Activity in Lesson 2 can be done as a homework or extra credit assignment.

• Lesson 1 (60 minutes): PPT lecture (30 minutes) Worksheet (30 minutes)
• Lesson 2 (60 minutes): PPT lecture (10 minutes) Worksheet (10 minutes) Microsoft Excel Activity (40 minutes)

Standards Addressed:

NGSS.ECOSYSTEMS:INTERACTIONS, ENERGY, AND DYNAMICS.HS-L-S2-2Use mathematical representations to support and revise explanations based on evidence about factors affecting biodiversity and populations in ecosystems of different scales

CCSS.MATH.CONTENT.HSF.IF.B.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

CCSS.MATH.CONTENT.HSF.IF.C.7.E Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Background information for Educators:

Lesson 1: Rate of change, exponential growth and logarithms.

All organisms grow as they acquire resources and convert it into body tissue. Ontogeny is the branch of biology that studies the origination and development of organisms. When you plot mass (growth) over time, you can see many different mathematical relationships. Some species (such as humans) grow until they reach a certain size and then they plateau. This is known as determinate growth. Other animals (such as fish and reptiles) have the potential for continual growth, known as indeterminate growth. Growth curves can have different shapes depending on how the rate of growth changes over an animal’s lifespan. For example, humans grow quickly as babies, and then slower later in life until they reach final adult size. Other animals grow linearly or exponentially over time.

Exponential or (geometric growth) is common function seen in many biological systems. Logarithms, which are the inverse of exponents, are a helpful way to simplify calculations with exponential functions. In particular, natural logarithms (ln) which use a base of e (euler’s number) are particularly important in biological systems. In an exponential growth function, the rate of change increases (or decreases) by a constant proportion. Relative growth rate (RGR) can be calculated by taking the natural logarithm of mass at two time periods, and dividing by time. Doubling time, or the amount of time for a quantity to double, can be calculated by taking the natural logarithm of 2 and dividing by the relative growth rate.

RGR = ln(mass 2) – ln (mass 1)

Time 2 – Time 1

Doubling time = ln (2)

RGR

Lesson 2: Consequences of climate change: temperature effects on growth rate

As consumers, all animals require energy to support the thousands of biochemical reactions necessary to sustain life. The sum of energy flowing through an organism over given period of time is known as its metabolic rate. An animal’s metabolic rate can change as conditions in the environment such as food quality or temperature change. Some animals such as birds and mammals are endotherms that use metabolic energy to generate heat and sustain their bodies at high temperatures, regardless of the environmental conditions. Other animals, such as invertebrates are ectotherms who do not use metabolic energy to heat their bodies, therefore their body temperatures are highly dependent on environmental conditions.

Animals use a variety of strategies to survive in cold places such as the Arctic. Many small organisms (such as mosquitoes) have an annual life cycle. They overwinter as eggs and hatch in the spring right after the ice melts on the ponds where they live. After living in ponds for 2-3 weeks, they emerge as adults, the females seek a blood meal, and then lay their eggs which remain dormant until the next spring. Other large mammals, such a muskox, have thick fur and a lot of body fat which helps them survive through the Arctic winter. As large herbivores (plant-eaters) they spend most of the spring and summer eating grasses, willows, lichens, and mosses.

As ectotherms, insects are highly sensitive to temperature change, and we are already starting to see the effects of climate change on insects around the world. One effect of increasing temperature is that many insects experience lower mortality rates (higher survival). Southern pine beetles (Dendroctonus frontalis), for example, are now becoming a problem in New Jersey and Long Island (and soon New England!) because of increasing winter minimum temperatures. Since the 2000s, winter minimum temperatures rarely reach -20° C, which is the critical temperature that kills of 50-90% of southern pine beetles. Another consequence of climate change on insects is that many are expanding their geographic range. One example of this is the spread of the Asian tiger mosquito (Aedesalbopictus) in the Southern USA. This is a concern of humans this mosquito is an effective vector of diseases such as Zika virus. Finally, insects have faster metabolic rates in warmer temperatures.

Lepidoptera larvae (caterpillars) exhibit exponential growth until they pupate. Because they are ectotherms, the shape of their growth curve is highly sensitive to thermal conditions (they grow faster at warmer temperatures). In the Arctic (and other places in the world) there are caterpillar species that experience population outbreaks. During these outbreaks, they can consume an enormous amount of plant material and leave behind barren landscapes. Some scientists have suggested that a caterpillar outbreak may have contributed to Viking extirpation from Southern Greenland (they ate their crops!).

Now that we understand some of the effects of temperature on insects, we can ask the question: how will temperature affect caterpillar growth rate? We will use data from an experiment in Greenland where students collected caterpillar from shrubs, and then reared them at two different temperatures. We will plot the data in Microsoft Excel and use basic mathematical models to solve for relative growth rate and doubling time (see Lesson 1).

Lesson 1 Materials:

-PPT1: Rate of change, Exponential growth, and Logarithms

-Worksheet 1

Lesson 2 Materials:

-PPT2: Consequences of Climate Change: Temperature effects on growth rate

-Youtube video on Arctic Caterpillars

-Worksheet 2

-Excel Activity Worksheet

-Excel spreadsheet (completed)

-Excel spreadsheet (blank)

Worksheet 1: Growth functions, Exponents, and Logarithms

1. Define the following terms:

Ontogeny:

Determinate growth:

Indeterminate growth:

1. Draw a growth curve for the following organisms (Humans, Trees, Sharks)

1. Describe the shape of each function (Linear, exponential, decelerating, etc.) and circle whether the organisms shows determinate or indeterminate growth:

Shape: Shape: Shape:

Determine/Indeterminate Determine/Indeterminate Determine/Indeterminate

1. Practice with exponents and logarithms. (Use a scientific or graphing calculator)

1) log10(1000) =

2) e4 =

3) loge(30) =

4) 84=

5) ex = 65, x =

6) log10(x) = 5, x =

1. We fed leaves to caterpillars and measured their mass over time. Plot the data.

1. Use this formula to calculate Relative growth rate (RGR) from day 1 to day 6

RGR = ln(mass day 6) – ln (mass day 6)

Day 6 – Day 1

1. Use this formula to calculate Doubling Time (DT)

Doubling time = ln (2)

RGR

Worksheet 2: Arctic environmental change and insect growth rates

1. Define the following terms:

Endotherm:

Ectotherm:

Metabolism:

1. How does insect metabolism (relative performance) respond to temperature change? Sketch the relationship on this graph.

1. Give an example of an ectotherm and an endotherm and describe how they survive in cold places.

1. Describe three ways that insects are responding to increasing global temperatures and give an example for each.

1)

2)

3)

1. Fill in this graph below showing caterpillar mass over time. Assume the caterpillar lives for 5 days, and then pupates (metamorphizes into a moth or butterfly). Describe the shape of the function.

1. Assume the curve you drew above was for a caterpillar growing at 10° C. Now draw another curve on the same graph depicting a caterpillar growing at 18° C. Label each curve.

1. Write a formal hypothesis describing how caterpillar relative growth rate and doubling time will change (or not change) due to temperature.

1. What other aspects of the environment are likely to influence caterpillar growth rate? Design a simple experiment you could do to test another question about caterpillar growth rates (other than temperature).

Caterpillar Growth Rate Excel Activity:

1. Open the Caterpillar growth rate spreadsheet

1. Go to the first tab “Practice” and complete the Practice problems using Microsoft excel formulas.

1. Read over the Metadata tab

1. Go to the Raw Data tab. Fill in columns E, H, K, N, and Q by calculating the natural log (ln) of mass for each of the 5 time periods. Type the formula in cells E2, H2, K2, and Q2 and drag down the formula to the bottom of the column.

1. Calculate average mass and the natural log (ln) of the average mass for each of the temperature treatments over each of the 5 time periods (Mass.0 – Mass.4).

1. Calculate standard deviation for the average mass and natural log of the average mass for each of the temperature treatments over each of the 5 time periods (Mass.0 – Mass.4) using the “=stdev()” formula

1. When you are done with calculations on the “raw data” tab, go to the “Graph growth over time” tab. The values you calculated in the “raw data” tab should automatically appear in this new table.
2. Look at the graphs for Caterpillar growth rate over time. How do they differ? Do the data match your prediction?

1. Go the “RGR Doubling Time” Tab. Calculate the RGR by taking the difference in mass divided by the total number of days of the experiment.

1. Calculate Doubling Time by taking the natural log (ln) of 2 and dividing by the RGR.

1. Look at the graphs. Do the results support your prediction?