o o o oAmong
ooo
Episode: Invaders
Uso o o o o o o o o o
EXPLORING NORTH CAROLINA
Calculating Exponential Growth of Knapweed
STANDARD COURSE OF STUDY CORRELATIONS:
Science, Grade 6, 7, 8, Goal 1: The learner will design and conduct investi­
gations to demonstrate an understanding of scientific inquiry.
1.05 Analyze evidence to explain observations, make inferences
and predictions, and develop the relationship between evidence
and explanation.
1.06 Use mathematics to gather, organize, and present quantitative
data resulting from scientific investigations.
1.08 Use oral and written language to communicate findings.
1.09 Use technologies and information systems to research, gather and
analyze data, visualize data, and disseminate findings to others.
MATERIALS
o Photos of spotted knap­
weed (a large laminated
image or several smaller
photos to pass around).
An Internet search will pro­
duce many suitable images.
o Graph paper, one sheet
per student or pair
o Calculators (optional)
Math, Grade 6, Goal 1: The learner will understand and compute with rational numbers.
1.04 Develop fluency in addition, subtraction, multiplication, and division of non­
negative rational numbers.
• Analyze computational strategies.
• Describe the effect of operations on size.
• Estimate the results of computations.
• Judge the reasonableness of solutions.
1.05 Develop fluency in the use of factors, multiples, exponential notation, and
prime factorization.
1.06 Use exponential, scientific, and calculator notation to write very large and
very small numbers.
1.07 Develop flexibility in solving problems by selecting strategies and using mental
computation, estimation, calculators or computers, and paper and pencil.
Science, Grade 6, Goal 7: The learner will conduct investigations and use technolo­
gies and information systems to build an understanding of population dynamics.
7.01 Describe ways in which organisms interact with each other and with non-living
parts of the environment:
• Coexistence/Cooperation/Competition.
• Symbiosis.
• Mutual dependence.
7.02 Investigate factors that determine the growth and survival of organisms.
• Light.
• Temperature range.
• Mineral availability.
• Soil/rock type.
• Water.
• Energy.
7.05 Examine evidence that overpopulation by any species impacts the environment.
7.06 Investigate processes which, operating over long periods of time, have resulted
in the diversity of plant and animal life present today:
• Natural selection.
• Adaptation.
INTRODUCTION TO LESSON: Students will work individually or in pairs to calculate
the exponential growth of an invasive weed.
For a follow-along viewing guide for students, see Viewing Guide 15. Spotted knapweed photo courtesy of Joseph M. DiTomaso, University of California, Bugwood.org.
•1•
Calculating Exponential Growth of Knapweed
ooooooooo
Spotted knapweed infestation photograph by John M. Randall, The Nature Conservancy, Bugwood.org.
BACKGROUND FOR TEACHER: A non-native species becomes invasive when it domi­
nates, displaces and disrupts native species. When a non-native species is intro­
duced into habitats outside its native home, sufficient barriers to limit its growth
may not exist. Natural controls include predators, disease, floods, fires and hurri­
canes. Human control measures include habitat destruction, collecting/harvest and
pesticide application. When a plant species is capable of exponential growth due to
the quantity of seeds that it can produce, it can become an overwhelming problem
in a non-native habitat.
Spotted knapweed is an invasive plant that arrived in North America from Europe
and Asia in the late 1890s as a contaminant in alfalfa and clover seed. It has spread
throughout Canada and almost all of the United States (except Alaska, Georgia,
Mississippi, Oklahoma and Texas) and runs rampant in fields, forests, prairies,
meadows, pastures and rangelands. Spotted knapweed outcompetes native plant
species, reduces native plant and animal biodiversity and decreases forage produc­
tion for livestock and wildlife. It has increased at an estimated rate of 27 percent
per year since 1920 and has the potential to invade about half of all the rangeland
(35 million acres) in the state of Montana alone.
engage f
Ask students what they think prevents a species of plant or animal
from becoming overpopulated. Have groups of students discuss and compile their
ideas on paper.
explore f Show Chapter 4 of the video. Review with students how unchecked
growth of non-native organisms can overwhelm ecosystems. Show a photo of spotted
knapweed and talk about its natural history and the effects it has had on habitats in
the United States. Tell students that they will be calculating the potential exponen­
tial growth rate of one knapweed over a 10-year period. Have students work individually or in pairs to complete the assignment sheet. Assist as necessary.
•2•
BEYOND THE CLASSROOM
• Visit a state park to learn
about invasive species. Most
state parks can offer a topicspecific program. The N.C.
State Parks system (http://
www.ncparks.gov/) offers
free Environmental Education
Learning Experiences (EELEs)
for educators and students
that may be used before,
after, during and/or instead
of field trips (download from
http://www.ncparks.gov/
Education/eele.php). Park
staff will provide guided onsite activities with school
groups (call ahead to make
a reservation). The Parks
system has a stewardship
program focusing specific­
ally on invasive species.
Check out www.ncparks.gov/
About/natural_resources_
stewardship.php for more
information.
• A nearby public garden may
be able to provide a topicspecific program upon request.
For a list of public gardens
throughout North Carolina,
visit http://www.ces.ncsu.
edu/depts/hort/consumer/
pgpages.html.
Additional Resources:
Spotted Knapweed Fact Sheet
• http://www.nps.gov/plants/
ALIEN/fact/cest1.htm
Calculating Exponential Growth of Knapweed
Solution:
• From 100 seeds, four (4 percent) will germinate the first year.
• Ninety-six seeds will remain in the “seed bank” (stores of seed in the soil that have not germinated yet).
• Twenty-five percent of the sprouts (one of the four) will survive to maturity. That plant will produce
1,000 seeds the next year.
• In the second year, 44 seeds (4 percent of 1,096) in the seed bank will sprout; 11 of those plants will
survive, increasing the population to 12 plants (with first year plant added).
• These 12 plants produce 12,000 new seeds. Add 96 percent of last year’s seed bank for a total of
13,052 seeds.
• Continue counting for five years, taking 1 percent of the previous year’s seed bank and adding to the
previous year’s plants to calculate the number of new plants.
• Multiply plants by 1,000 and add 96 percent of the previous year’s seed bank to determine the new
size of the seed bank.
• Since the plants live for only five years, subtract Year 1 from Year 6, Year 2 from Year 7, and so on.
• Seeds are viable for only eight years, so subtract 100 from Year 8’s seed bank, Year 1’s seed bank
from Year 9’s seed bank, and so on.
*Under ideal conditions, knapweed could reproduce at this rate. Many variables, including soil type and
precipitation, may affect population growth.
Year
0
1
2
3
4
5
6
7
8
9
10
Number of Plants
0
1
12
143
1,698
20,171
239,614
2,846,408
33,812,879
401,667,920
4,771,469,407
Number of Seeds
100
1,096
13,052
155,530
1,847,309
21,944,417
260,680,640
3,096,661,414
36,785,673,858
436,982,165,807
5,190,972,273,123
explain f Have students share their answers and the method(s) by which they arrived at the solutions.
Assist any students unable to do the math.
elaborate f Have students graph their results to illustrate the rate of growth over time.
evaluate f Show Chapter 2 of the video to introduce students to other species that have become invasive.
Ask students if the bird species mentioned (starlings, house sparrows, rock doves and house finches) might
grow in number as quickly as the knapweed. Have them explain in writing why or why not.
Activity adapted from the Montana Weed Project Teacher’s Handbook.
•3•
Calculating Exponential Growth of Knapweed
Assignment
Calculate the exponential growth of knapweed.
One hundred knapweed seeds are inadvertently dropped in an area. Calculate
how many plants and seeds will be produced over a period of 10 years based
on the following growth factors:
• One knapweed plant produces 1,000 seeds.
• Four percent of knapweed seeds germinate each year, leaving 96 percent
for the following year.
• About 25 percent of seedlings will survive to maturity.
• Knapweed seeds remain viable (able to germinate) in the soil for eight years.
• The time from germination to seed production is one year.
• Knapweed plants live for five years.
Year
0
Number of Plants
Number of Seeds
0
100
1
2
3
4
5
6
7
8
9
10
Illustration by Tracey Saxby, IAN Image Library (ian.umces.edu/imagelibrary/).
•4•