ALEX Classroom Resource

  

The Limits of Tolerance

  Classroom Resource Information  

Title:

The Limits of Tolerance

URL:

https://www.ck12.org/c/calculus/evaluate-limits-using-substitution/rwa/The-Limits-of-Tolerance/?referrer=concept_details

Content Source:

Other
CK-12
Type: Informational Material

Overview:

They're an important part of the ecosystem. They prevent disease and clean up carrion. Yet, they're also a nuisance to homeowners and a threat to livestock. Their population has recovered and grows at an incredible rate. At what point can we say that there are too many black vultures in America?

This informational material will apply a precalculus concept--limits of functions--to an environmental science issue--how biological and physical changes within an ecosystem can affect the population growth of a species. There are additional links provided for students to explore more about this issue. 

Content Standard(s):
Science
SC2015 (2015)
Grade: 9-12
Environmental Science
3 ) Use mathematics and graphic models to compare factors affecting biodiversity and populations in ecosystems.

Unpacked Content
Scientific And Engineering Practices:
Developing and Using Models
Crosscutting Concepts: Scale, Proportion, and Quantity
Disciplinary Core Idea: Earth and Human Activity
Evidence Of Student Attainment:
Students:
  • Use mathematical and/or graphical representations to compare factors affecting populations in an ecosystem.
  • Use mathematical and/or graphical representations to compare factors affecting biodiversity in ecosystems.
  • Compare the effects of limiting factors on biodiversity and populations in ecosystems.
Teacher Vocabulary:
  • interpolation
  • extrapolation
  • anthropogenic
  • limiting factors
  • biodiversity index
  • species richness
  • species evenness
  • population
  • graphic models
  • population pyramid
  • doubling time
  • growth rate
  • slope
  • exponential growth
  • population curve
  • logistic growth model
  • linear growth model
  • constant growth
  • density-dependent limiting factors
  • density-independent limiting factors
  • carrying capacity
  • Biodiversity Treaty
  • demographic transition
  • correlation
  • endangered species
  • extinction
  • survivorship
  • sustainability
  • population properties
  • density and dispersion
  • reproductive potential
Knowledge:
Students know:
  • The carrying capacity of an ecosystem results from such factors as availability of living and nonliving resources and from such challenges as predation, competition, and disease.
  • Anthropogenic changes in the environment, including habitat destruction, pollution, introduction of invasive species, overexploitation, and climate change, can disrupt an ecosystem and threaten the survival of some species.
  • Examples of mathematical representations include finding the average, determining trends, and using graphical comparisons of multiple sets of data.
  • The difference between density-dependent and density-independent limiting factors, examples of each, and how each affects populations and biodiversity within an ecosystem.
Skills:
Students are able to:
  • Differentiate between constant and exponential growth.
  • Use graphs to compare multiple sets of data.
  • Determine trends in data sets.
  • Use a variety of graphs and charts, including: (e.g., scatterplots, tables, line graphs, bar graphs, histograms) to evaluate the impact of factors on populations and biodiversity.
  • Utilize interpolation, extrapolation and statistical analyses to determine relationships between biodiversity and population numbers.
  • Make inferences and justify conclusions from sample surveys, experiments, and observational studies. (ALCOS Mathematics S-IC)
  • Choose a scale and the origins in graphs (ALCOS Mathematics ALGI. 4.2) in order to accurately compare graphical data.
  • Determine an appropriate graphic model to display relationships comparing populations by biodiversity.
  • Describe how factors affecting ecosystems at one scale can cause observable changes in ecosystems at a different scale.
Understanding:
Students understand that:
  • The number of populations in a given area reflects the biodiversity of that area.
  • Ecosystems can exist in the same location on a variety of scales, and these populations can interact in ways that may, or may not, significantly alter the ecosystems.
  • Using the concept of orders of magnitude, a model at one scale relates to a model at another scale.
AMSTI Resources:
ASIM Activities include:
Exponential Population Growth
Science
SC2015 (2015)
Grade: 9-12
Environmental Science
4 ) Engage in argument from evidence to evaluate how biological or physical changes within ecosystems (e.g., ecological succession, seasonal flooding, volcanic eruptions) affect the number and types of organisms, and that changing conditions may result in a new or altered ecosystem.


NAEP Framework
NAEP Statement::
L12.7: Although the interrelationships and interdependence of organisms may generate biological communities in ecosystems that are stable for hundreds or thousands of years, ecosystems always change when climate changes or when one or more new species appear as a result of migration or local evolution. The impact of the human species has major consequences for other species.


Unpacked Content
Scientific And Engineering Practices:
Engaging in Argument from Evidence
Crosscutting Concepts: Cause and Effect
Disciplinary Core Idea: Earth and Human Activity
Evidence Of Student Attainment:
Students:
  • From the given explanation, identify the claims to be evaluated, the evidence to be evaluated, and the reasoning to be evaluated.
  • Evaluate, based on evidence, how biological changes within ecosystems affect the number and types of organisms.
  • Evaluate, based on evidence, how physical changes within ecosystems affect the number and types of organisms.
  • Engage in argument from evidence to assess how changing conditions may result in a new or altered ecosystem.
Teacher Vocabulary:
  • ecological succession
  • seasonal flooding
  • volcanic eruptions
  • ecosystem
  • biological changes
  • physical changes
  • keystone species
  • pioneer species
  • habitat alteration
  • density-dependent limiting factors
  • density-independent limiting factors
  • primary succession
  • secondary succession
  • remediation/bioremediation
  • symbiosis
  • abiotic factors
  • biotic factors
  • food chain
  • food web
  • energy pyramid
  • energy flow
  • bioaccumulation
  • ecological system
  • ecosystem services
  • deforestation
  • organism
  • species
  • population
  • community
  • ecosystem
  • biome
  • biosphere
  • desertification
  • overharvesting
  • overgrazing
  • pathogen
  • climax community
Knowledge:
Students know:
  • The components of a scientific argument including the claim, alternative claim, evidence, justification, and the challenge to the alternative claim.
  • Factors that affect biodiversity.
  • The relationships between species and the physical environment in an ecosystem.
  • Examples of biological changes (e.g., ecological succession, disease) and physical changes (e.g., volcanic activity, desertification) that affect the number and types of organisms, and that may result in a new or altered ecosystem.
Skills:
Students are able to:
  • Use additional relevant evidence to assess the validity and reliability of the given evidence and its ability to support the proposed argument.
  • Describe the strengths and weaknesses of the given claim in accurately explaining a particular response of the ecosystem to a changing condition, based on an understanding of factors that affect biodiversity and the relationships between species and the physical environment.
  • Assess the logic of the reasoning, including the relationship between degree of change and stability in ecosystems, and the utility of the reasoning in supporting the explanation.
Understanding:
Students understand that:
  • A complex set of interactions within an ecosystem can keep its numbers and types of organisms relatively constant over long periods of time under stable conditions.
  • When modest biological or physical disturbances occur in an ecosystem, it returns more or less to its original status (i.e., it is resilient).
  • Extreme fluctuations in conditions or the size of any population, however, can challenge the functioning of an ecosystem in terms of resources and habitat availability, and can even result in a new ecosystem.
AMSTI Resources:
ASIM Activities include:
Predator-Prey Populations; Bluegill Limiting Factors; Limiting Factors; Bio-Assessment; Changes in an Ecosystem

Alabama Alternate Achievement Standards
AAS Standard:
SCI.AAS.ES.HS.4- Recognize changes within ecosystems that affect the number and types of organisms in that ecosystem.


Mathematics
MA2019 (2019)
Grade: 9-12
Precalculus
7. Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity.

a. Apply limits of functions at specific values and at infinity in problems involving convergence and divergence.

Unpacked Content
English Language Arts
ELA2021 (2021)
Grade: 11
8. Read, analyze, and evaluate texts from science, social studies, and other academic disciplines and explain how those disciplines treat domain-specific vocabulary and content and organize information.
Unpacked Content
Key Academic Terms:
8.
  • Academic disciplines
  • Domain-specific vocabulary
  • Content organization
Knowledge:
8. Students know:
  • Strategies to read, analyze, and evaluate texts from various academic disciplines.
  • Content-specific text will often include a particular structure and domain-specific vocabulary.
Skills:
8. Students are able to:
  • Read, analyze, and evaluate texts from subjects other than English language arts to determine the use of domain-specific vocabulary.
  • Read, analyze, and evaluate texts from subjects other than English language arts to determine how the academic discipline organizes content.
Understanding:
8. Students understand that:
  • Different academic disciplines may utilize different vocabulary.
  • Different academic disciplines may arrange content in particular organizational styles.
English Language Arts
ELA2021 (2021)
Grade: 12
8. Read, analyze, and evaluate texts from science, social studies, and other academic disciplines and explain how those disciplines treat domain-specific vocabulary and content and organize information.
Unpacked Content
Key Academic Terms:
8.
  • Academic disciplines
  • Domain-specific vocabulary
  • Content organization
Knowledge:
8. Students know:
  • Strategies to read, analyze, and evaluate texts from various academic disciplines.
  • Content-specific text will often include a particular structure and domain-specific vocabulary.
Skills:
8. Students are able to:
  • Read, analyze, and evaluate texts from subjects other than English language arts to determine the use of domain-specific vocabulary.
  • Read, analyze, and evaluate texts from subjects other than English language arts to determine how the academic discipline organizes content.
Understanding:
8. Students understand that:
  • Different academic disciplines may utilize different vocabulary.
  • Different academic disciplines may arrange content in particular organizational styles.
Tags: algebra, algebraically, analytically, asymptote, ecosystem, end behavior, function, graph, hole, limit, limit notation, population, precalculus
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  This resource provided by:  
Author: Hannah Bradley