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Home Up Scale and Mapping Invasion rates Quadrat sampling Graphing 1 Graphing 2

“The Plot Thickens”:  Using Quadrats to Estimate Abundance and Densities of Organisms

Grade Level: 6-8

Time Frame: 2 - 3 class periods

Subject:  Math

Introduction to Lesson: 

In this lesson students will continue to review the concept of using symbols to represent real-world objects or organisms.  However, the focus of the application is now on using quadrat estimation to count the numbers of individuals within a population.  Students will also work with enumerating the number of kinds of organisms, estimating their abundance and density, as well as reviewing the use of percentages in the context of community composition.  They will use these skills to assess the effects of a Phragmites  invasion into a hypothetical ecosystem.

New Jersey Core Curriculum Content Standards

STANDARD 4.1 (Number and numerical operations) All students will develop number sense and will perform standard numerical operations and estimations on all types of numbers in a variety of ways.

A. Number Sense

·          Use real-life experiences, physical materials, and technology to construct meanings for numbers 

o         Explore the use of ratios and proportions in a variety of situations.

C.  Estimation

·          Use a variety of strategies for estimating both quantities and the results of computations.

·          Recognize when an estimate is appropriate, and understand the usefulness of an estimate as distinct from an exact answer.

·          Determine the reasonableness of an answer by estimating the result of operations.

·          Determine whether a given estimate is an overestimate or an underestimate.

STANDARD 4.2 (Geometry and measurement) All students will develop spatial sense and the ability to use geometric properties, relationships, and measurement to model, describe and analyze phenomena.

D. Units of Measurement

·          Use measurements and estimates to describe and compare phenomena

E. Measuring Geometric Objects

·          Develop informal ways of approximating the measures of familiar objects (e.g., use a grid to approximate the area of the bottom of one's foot).

STANDARD 4.4 (Data analysis, probability, and discrete mathematics) All students will develop an understanding of the concepts and techniques of data analysis, probability, and discrete mathematics, and will use them to model situations, solve problems, and analyze and draw appropriate inferences from data.

A. Data Analysis (or Statistics)

·          Collect, generate, organize, and display data.

·          Read, interpret, select, construct, analyze, generate questions about, and draw inferences from displays of data.

·          Range, median, and mean

·          Calculators and computers used to record and process information

·          Respond to questions about data, generate their own questions and hypotheses, and formulate strategies for answering their questions and testing their hypotheses.

STANDARD 3.1 (Reading) All students will understand and apply the knowledge of sounds, letters, and words in written English to become independent and fluent readers, and will read a variety of materials and texts with fluency and comprehension.

A. Concepts About Print/Text

Use a glossary independently and appropriately.

C. Decoding and Word Recognition

Apply knowledge of new words correctly

STANDARD 3.5 (Viewing and media literacy) All students will access, view, evaluate, and respond to print, nonprint, and electronic texts and resources.

Constructing Meaning

Respond to and evaluate the use of illustrations to support text.

Identify the central theme in a movie, film, or illustration.


bulletStudents will use a quadrat survey technique to analyze the biodiversity of a model marsh community
bulletStudents will use the quadrat technique to assess changes in native species abundance and diversity as a result of that marsh being invaded by Phragmites.
bulletStudents will practice calculation of averages (computing the mean) in the context of the data collected from the model Phragmites  marshes
bulletStudents will  review the concept of density, and to apply that concept to the calculation of the densities of the various plant types within the model marsh
bulletStudents will answer discussion questions about what these numbers mean in terms of the impact of the Phragmites invasion for the plants and animals of marshes

Materials and Resources:

bulletPosters (caution.  Large files especially time 3 and 4.  Some assembly required.... print on regular printer and then stick together with tape to assemble poster)
bulletMarsh, time 1 (publisher formatpdf format)
bulletMarsh time 2 (publisher formatpdf format)
bulletMarsh time 3 (publisher formatpdf format)
bulletMarsh, time 4 (publisher formatpdf format)
bullet"Quadrats" (template in word format;  template in pdf format)
bulletQuadrat student worksheet.  Quadrat Datasheet


bulletQuadrat:  A set area (often a square or rectangle) that ecologists use to isolate a sample area within which the diversity and abundance of organisms (e.g. plants, fungi, large algae and some kinds of relatively sedentary [stationary or slow moving] animals) can be counted. Also used to describe the square- or rectangle-shaped measuring device used to establish a quadrat.  
bulletEstimate:  an approximation of the value or amount of something
bulletReplicate:   A repeated sample of a similar quantity taken in the same way as other “replicates” from a larger population. Replicates are used in science because a single sample may not deliver an accurate picture of what is “typical” for the larger population.  Thus, multiple samples are used to try to “even out” the differences between samples taken from the larger population so that a more accurate picture of the characteristics of the larger population can be generated.
bulletExtrapolation (verb: to extrapolate):  Using knowledge of trends and patterns within a data set to make predictions for situations beyond those within the measured data set.  For example, a child has been growing half an inch a month for the last 6 months.  We can thus predict that, in two months from today, that child will be 1 inch taller than she is today.
bulletMean (Average):  One way to estimate a typical value for a data set.  Also known as a “mean” value 
bulletMode:  The most common (most frequently observed) value within a data set
bulletMedian:  The value of the data point that lies in the middle of a given data set.  In an ordered data set, half the data will be larger than value and the other half will be smaller than it.
bulletRange:  The difference between highest and lowest observed values in a data set.  It can be expressed as an interval, e.g. 15-27 where 15 is the lowest observed value and 27 is the highest.  It can also be expressed as the interval width, so for our example the range 15-27 can also be written as a range of 12.
bulletDensity:  This word has several meanings.  In the context of this lesson density refers to a number that describes the number of something (e.g. number of a type of plant, or animal) present within a given area. 
bullet Diversity:   The number of different KINDS of things present within a given area


Anticipatory Set: 

Show students the PowerPoint slides of marshes (provided). Engage students in a discussion of what they see. Guide them to observe differences in (1) the number of different kinds of plants present and (2) the number of plants present (density). Ask them to imagine that they’ve been assigned to find out how many of each type of plant is present within each of the marshes.  How long do they think it would take them to count all of the plants of each type within the marsh?  Once it becomes clear that there are too many to count easily, ask the students “how else could they find the answer?”  Hopefully the students will come up (can be guided to) the idea of counting smaller “representative” areas and then “extrapolating” from those to the larger marsh.  The teacher can then show students the final two slides in which scientists are shown carrying a quadrat to a marsh and then using it to count the different kinds of plants within the designated area.  Point out that the scientists must first identify the plant types and then count (enumerate) the number of each type within the area delineated by the quadrat.  The teacher will then lead the students to a discussion on how, once one has an estimate of the numbers and types of organisms within a set of quadrats, these numbers could be used to develop and estimate an estimate of what are the typical (or “average”) values for the numbers and types of species in this marsh?  (Hopefully once the students hear the substitution of the word “average” for the word “typical” this will prompt they to offer the idea of computing an average or mean for each plant type / quadrat.  Finally, ask the students if they know the average number of plants per quadrat for a marsh, and the marsh’s area is 100x that in the quadrat, how many plants would they expect (approximately) would there be in the whole marsh.  Then ask, if the marsh’s area was some other number of times larger than that sampled within the quadrat, how would they work out how many plants (total) were in the marsh?  Once the students have seen the pattern that the total number of plants in the marsh can be approximated by working out what fraction of the marsh’s area is represented within the quadrat and then multiplying the “typical” or average number of plants within the quadrat by that amount, they can estimate the total number of each plant type within the larger marsh area.

Sequence Instruction:

1.       Each group of students will be provided with a set of posters representing a marsh in New Jersey and a set of “quadrats” (transparent plastic squares with cardboard frames). 

2.       Students will be asked to locate the legends.  The teacher will then review with the students the type of plant represented by each symbol.

3.       Students will then be given two minutes to brainstorm in their group for a good way to place their quadrats onto the “map” in such a way as to create representative “replicate” measurements of the kinds of plants within the marsh.  Ideal strategies would involve either some degree of randomness (e.g. throwing the quadrat with one’s eyes closed), or strategies that employ a variant of a “systematic” sampling design (e.g. (1, 1), (2,2), (3,3) within an imagined grid system superimposed on the marsh, where (1, 1) would be 1 quadrat unit from the left and 1 quadrat unit from the top).

4.       Students will be asked to share their group’s strategy with their classmates.  This will allow the teacher to assess the suitability of the suggested strategies, as well as providing groups who had struggled with this concept with guidance on how to achieve this.  With the help of the teacher the class will either pick a single strategy or agree on a suite of acceptable strategies which can be used in sampling.  Ideally students in one group should NOT sample exactly the same plots within each marsh as those in other groups sampling the same marsh, since later on we want students to see that they all get slightly different answers to the same questions as a result of the locations of their subset of quadrats.  This isn’t a problem if a random sampling design is employed, but if students use a systematic approach the teacher should encourage use of different patterns within each group so that a different subset of quadrats is counted within each marsh.

5.       Students will then make at least 10 replicate quadrat samples from their “native marshes” and enter the data into the table provided and make a variety of calculations (column averages, estimates of biodiversity)

6.       Students will then repeat this exercise for each model marsh (at different stages of invasion), recording the same data for each one.  (To save time the teacher may wish to assign different groups to collect data from marshes at different stages in the invasion.  However, there should be some overlap (more than one group assessing diversity and densities for each marsh) to allow later discussion of why different groups got different answers even though they had sampled the same marsh).

7.   The number of different kinds (species) of organisms within an area is known as biodiversity.  Students will assess the biodiversity of the various model marshes by counting the number of KINDS of plant present in each replicate quadrat for each model marsh assessed.

8.   Students will report their findings on biodiversity for each quadrat that they measured, and the teacher will tally the results on the board or a chart so that all students can see the results from all groups.

9.   What is the range of values for biodiversity seen within quadrats in each model marsh?

10.   What is the mode for the data set on biodiversity in the quadrat samples from each marsh model?

11.   What is the median number of kinds (biodiversity) that was seen in the quadrats from each marsh model?

12.   Students will compute the mean (average) of their counts of the number of kinds (biodiversity) of plants within each quadrat for each model marsh assessed. 

13.   The students will then tally the number (abundance) of individuals of each kind of plant within each quadrat for each of their assigned model marshes

14.   Students will calculate density (number of stems per square meter) for different plants within their quadrats  for each of their assigned model marshes

15.    Students will calculate the average density of each plant type within the quadrats sampled for each of their assigned model marshes.

16.   Students will calculate the fraction and percentage of each kind of plant within the community for each of their assigned model marshes

17.   Students will calculate the total area of the marsh (the poster) 

18.   Students will calculate the ratio of their quadrat size to that of the total size of the marsh (i.e. what fraction of the marsh area is represented within one quadrat?)

19.   Students will use this information to extrapolate from their average counts for each plant type quadrat samples to the overall marsh in order to calculate how many plants of each species and in total are present in the marsh at each stage in the invasion

20.   Students will be asked to select a “spokesperson” who will report their groups’ calculated densities for several key species (requested by the teacher) at a given stage within the invasion to the class.   Teacher will tally these results on the board, chart or similar so that the answers for the different groups can be seen side by side for the different species and stages of invasion requested. Assuming all went well, the estimates provided from the different groups should be similar, but often will not be identical.  Encourage students to discuss why this might be and what this means for the accuracy of their estimates.  In particular, students may observe that variability is particularly high for relatively rare species.  Encourage students to speculate on why this might be.

Note:  Data from the quadrat samples taken in this lesson plan will be used again in several succeeding lessons.  Remind students to put the data somewhere safe and to be sure that they bring it to all subsequent classes.  (Teacher may want to collect copies of all data generated in this class, just in case it gets lost and needs to be provided to the students again when the next need it).

Accommodations and Modification:                                                                        

·        Group students into groups such that each group has a mix of students with different learning styles and strengths.

·        Group students such that each group has a mix of students with different learning styles and strengths.

·        Provide students with visual impairment with alternative visual media (printouts of PowerPoint slides, graphs made with puffy pens or embossing) to allow them to participate in the games

·        If needed, pair students so that one student can read directions aloud to another

·        Provide handouts and worksheets with larger font sizes, raised or embossed writing, or Braille for visually impaired students

Closure   / Homework

Have students discuss within their groups and then report back to the class on one of the following questions  (teacher should assign questions to groups).  Alternately, have students write a paragraph on one or more of the following for homework.

  1. What would be the advantages and disadvantages of using a larger / smaller quadrat to survey the marsh?
  2. What were the main trends in terms of diversity, number of samples containing Phragmites, Phragmites density, and overall increase in stem density over time?

3.       What do you think the marsh would look like if you could visit it and walk around in it at different times within the invasion?  What would you see?  How hard would it be to move through the vegetation in the marsh (don’t worry about sinking in the mud… you have special shoes on that prevent this.  We’re just talking about how it would be to move through the plants here)?

4.       What do you think the marsh would feel like to an animal that fed on / nested in one of the native plants present before the invasion as the Phragmites invasion takes place?   What would this mean for those animals?

5.       If reinforcement of graphing formats is desired, provide students with additional data sets and ask them

a.       What is the appropriate graph format for data of that type and why?

b.       Create the graph for each data set using the format that you’ve identified.


TEACHER FEEDBACK REQUEST:  We are always to working to improve these lesson plans. If you use this lesson plan, we'd love to hear from you with your thoughts, comments and suggestions for future improvements.  Please take the time to fill in our survey at .  Thanks!


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© 2009. Louise Wootton.  Edited by Claire Gallagher 

 Although the information in this document has been funded wholly or in part by the United States Environmental Protection Agency under assistance agreement NE97262206  to Georgian Court   University, it has not gone through the Agency's publications review process and, therefore, may not necessarily reflect the views of the Agency and no official endorsement should be inferred.


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