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What speed the invasion?  Estimating the rate of invasion of Phragmites  in a Delaware Marsh

 Grade Level: 6-8 

Time Frame: 2 class periods

 Subject:  Math

 Introduction to Lesson:  In this lesson students will practice what they’ve learned about reading and interpreting maps and map legends through active use as part of solving problems pertaining to the invasion of Phragmites  in a Delaware marsh.  They will use the information that they obtain from reading the maps to calculate rates of expansion of the invasive Phragmites, first in a simplified model and then in a more realistic approximation.  In the process the students will review how to calculate the area and perimeter of geometric shapes (rectangle, ellipse).  They will also practice calculating percentages within the context of the Phragmites  invasion. 

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 

·          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 a scale to find a distance on a map or a length on a scale drawing.

·          Convert measurement units within a system (e.g., 3 feet = ___ inches).

·          Know approximate equivalents between the standard and metric systems (e.g., one kilometer is approximately 6/10 of a mile).

·          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 4.3 (Patterns and algebra) All students will represent and analyze relationships among variable quantities and solve problems involving patterns, functions, and algebraic concepts and processes

A. Patterns

·          Recognize, describe, extend, and create patterns involving whole numbers and rational numbers.

o        Descriptions using tables, verbal rules, simple equations, and graphs

C. Modeling

·          Use patterns, relations, and linear functions to model situations.

o        Using variables to represent unknown quantities

o        Using concrete materials, tables, graphs, verbal rules, algebraic expressions/equations/inequalities

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.

o        Data generated from surveys

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

o        Range, median, and mean

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

A. Constructing Meaning

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

Objectives: 

  1. Students will apply what they’ve learned about mapping and scale to solve real world problems
  2. Students will practice using measurements of distance and time to calculate rates
  3. Students will successfully extrapolate from an existing set of data to predict future trends.
  4. Students will practice calculations of areas of regular shapes (rectangle, oval) in order to solve real world problems
  5. Students will practice calculations of percent in the context of a real world problem
  6. Students will use critical thinking skills to assess patterns observed within a data set

Materials and Resources:

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Paper

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Rulers (must have metric units- i.e. cm and mm)

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Calculators

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Worksheets

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PowerPoint  (same in pdf format)

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Printouts of the PowerPoint  slides of “invaded” marsh

Anticipatory Set: 

Teacher will show students a PowerPoint presentation featuring 3 slides showing expansion of Phragmites within a Delaware marsh and will review concepts such as scale bar, legend and compass rose with students.  Teacher will use the slides to engage students in a discussion about what they notice about the changes in the area occupied by Phragmites over time.  Teacher will involve students in a discussion of the idea of rates and how to measure rates, and will have the students write a formula for how far (linear) a population of plants moves over time (change in distance over time).  Teacher will then lead students to develop a similar discussion of increase in area over time.  This may provide an opportunity to assess the students’ starting point in terms of  what they do or don’t remember / know about calculating areas of various shaped objects (square, rectangle, circle, oval etc.).  Finally (or later in the class when the topic comes up), teacher will discuss perimeter with students.  Students will be encouraged to verbalize and clarify the difference between perimeter (a linear measurement) and area.  Use of a physical example (e.g. measuring the dimensions of a text book or other similar item) to illustrate this idea may prove useful in clarifying this distinction within students’ minds.

Sequence Instruction:

 Using the pictures provided in the PowerPoint provided, in which invasion is approximated by rectangular shapes, students will complete the assigned worksheets.

  1. Use a ruler to measure the horizontal distance that the Phragmites moved into the hypothetical marsh between 1900 and 1920.   

  2. Use the scale bar to convert this distance to the meters that the reed moved in this period.

  3. How far did the Phragmites move per year, on average, between 1900 and 1920?

  4. Repeat this exercise for the distance that the “invasion front” moved between 1920 and 1945

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What measurement did you get with your ruler?

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What did this convert to in terms of the distance (in meters) that the reed moved in this period?

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How far did the Phragmites move per year, on average, between 1920 and 1945?  (Note:  when you calculate this, you are calculating a rate, which is defined as "distance divided by time".

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Was this faster or slower than it moved between 1900 and 1920?

  1. Repeat this exercise for the distance that the “invasion front” moved between 1945 and 1980

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What measurement did you get with your ruler?

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What did this convert to in terms of the distance (in meters) that the reed moved in this period?

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How far did the Phragmites move per year, on average, between 1945 and 1980?

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Was this faster or slower than it moved between 1900 and 1920?  Than between 1920 and 1945?

  1. Repeat this exercise one final time for the distance that the “invasion front” moved between 1980 and 2000.

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What measurement did you get with your ruler?

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What did this convert to in terms of the distance (in meters) that the reed moved in this period?

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How far did the Phragmites move per year, on average, between 1990 and 2000?

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Was this faster or slower than it moved between each of the previous periods?

  1. Within your class or group, brainstorm:  Why might a plant be expected to spread faster during some periods than others?  Share your group’s answers with the class.

  2. If the expansion of Phragmites continues at the rates seen between 1980 and 2000, how much longer will it take for the plant to invade the “whole marsh” (the full page of the diagram)?

  1. The area of a rectangle can be found by multiplying the length of the base (L) times its height (H).    (A = L * H)

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What is the area of the marsh invaded in 1920?

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What is the unit of area in this case?

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What is the area invaded in 1945?

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What is the area invaded in 1980?

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What is the area invaded in 2000?

  1. By what percent did the invaded area increase between 1920 and 1945?

  2. The area of marsh invaded in 1980 was what percentage less than that in 1980?

  3. The perimeter of a rectangle can be found by adding two times its height to two times the length of its base.  (P =  (2 * L) + (2 * H) or  P = 2 (L + H)

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What is the perimeter of the area of marsh invaded in 1920?

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What is the unit for perimeter in this case?

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What is the perimeter of the area of marsh invaded in 1945?

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What is the perimeter of the area of marsh invaded in 1980?

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What is the perimeter of the area of marsh invaded in 2000?

In the real world the shapes of the invaded area are not rectangles.  They can be better approximated by ovals.  In mathematics, an "oval" shape is referred to as an “ellipse”. As you may remember, the area of an ellipse can be calculated using the equation π*a*b, where π is pi (3.14….), “a” is half of the length of the longer axis of the oval (measured at its greatest width) and “b” is half of the length of the narrower side (measured at its greatest height). 

Use the provided maps in which ovals are used to approximate (estimate) the shapes of the populations of invasive Phragmites in Delaware’s Silver Run Marsh, which we looked at earlier, to answer the following questions:

  1. Your teacher will assign you (or your group) one or more ellipses representing the area occupied by a population of Phragmites in Silver Run Marsh in 1954. 

  2. Use a ruler to obtain the length and height of the image your assigned ellipse(s) in centimeters and milimeters.

  3. Use the information scale bar to convert those dimensions to kilometers.

  4. Given that a kilometer is 1000m, convert those dimensions into meters.

  5. Estimate the area of your assigned ellipse(s) using the formula provided.

  6. What are the units of area in this case?

  7. Look on the map of the same marsh in 1968 and find the population that has developed from the one(s) you measured in the image from 1954.   Carry out the same measurements and conversions to calculate the area of that bed in 1968.

  8. What was the total area (all ellipses) invaded in 1954?  In 1968? 

  9. By what percent has each population increased in time between 1954 and 1968?

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A few of the populations have merged between the two dates.  What problems did this present to you in making the calculations for this question?

  1. What is the rate (in m2/ year) of expansion for each population.

  2. Have all of your populations expanded by the same amount in this period?  If not, within your class or group brainstorm:  Why might a plant be expected to spread faster during some periods than others? 

  3. What problems do you encounter in trying to estimate how fast each population expanded in area between 1968 and 1989? 

  4. Brain Storm: How might you estimate the total area invaded in 1989, now that all of the beds have fused together?  (teacher may want to guide this discussion toward discovery of the idea of using grids and counting squares.) 

  5. Carry out your proposed method.  (Use transparencies with squares onto which the students can trace the outline of the yellow (invaded) areas.  Then have students count squares to estimate area.  Students should be guided to a discussion of estimation and what to do about squares that are partially invaded and partially not).

  6. What is your best estimate of the total area invaded? 

  7. What are some of the things that will have influenced the accuracy of your estimate?

  8. What do you think?  When someone says something is an estimate, can you rely on that number or not?

  9. By what percent has each population increased in time between 1968 and 1989?

  10. What is the rate (in m2/ year) of expansion for each population over this period?

  11. How does that compare with that seen in the period between 1954 and 1968?

  12. Calculating the perimeter (P) of an ellipse is actually very tricky.  It can’t be done 100% accurately without using calculus.  However, a couple of pretty good estimates are available.  Here are two of the better ones:

               
        Easier:    P =    ( (3A+3B) – sqrt ((A+3B)*(B+3A)) )
 
        Better but slightly more complicated is
               
                               P =  (a+b)(1 + 3x2/[10 + sqrt(4 - 3x2)])  
 
                                              Where x = (a-b) / (a+b).

a.       Use both of these to calculate the real world estimated perimeter length for your assigned population in 1954? 

b.      What is the perimeter length for that population in1968? 

Note:  Data on the areas, perimeters and rates developed in this lesson plan will be used again in 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 they next need it).

Accommodations and Modification:  

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Group students into groups such that each has a mix of students with different learning styles and strength

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Provide students with visual impairment with alternative visual media (printouts of PowerPoint slides) to allow them to participate in the games

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

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If needed, provide written instructions / closed captioning for students who are hard of hearing.

 

Assessment / Anchor Activity (if lesson goes shorter than planned)

To reinforce the distinction between the definition and formulae for perimeter and those for area, teachers may want to have students to walk the sides of common 2D shapes (square, rectangle, circle, ellipse) while unraveling a ball of string.  The length of the unwound string can then be measured using a yard / meter stick to help consolidate student’s understanding that perimeter is a measure of length.  Students can then be challenged to calculate the area of the same shape.  This may mean returning to the shape to measure additional information (specifically the diameter of circle in order to calculate radius, width and height of an elliptical shape).  Such an exercise also helps to reinforce students’ understanding of what a square foot, square meter, foot or meter actually looks like (real-world scale).

Closure

Provide students with two yellow stick-its.  Ask them to pick two of the following to complete the sentences in a way that they feel is meaningful for them.  Have them stick their stick-its on the door or other similar surface as they leave.  Use the feedback to see if there are ideas or concepts that need to be reviewed at the start of the next class or one on one with a specific student.

1. One thing I learned today was ...

2. Something from today's lesson that I want to find out more about is ...

3. Something I found difficult to understand today was ...

4. I really enjoyed today's lesson because ...

5. I found today's lesson difficult because.... 

 

Homework/ Practice questions

Bed Size, Time 1:  2000 (1 cm represents 1 km “on the ground”)

 

 

 

 

Bed Size:  Time 2:  2009  (1 cm represents 1 km “on the ground”)

 

 

 

 

 

 

1. What was the length and width of the bed (in km) in 2000?

Length

 

Width

 

2. What was the length and width of the bed (in km) in 2009?

Length

 

Width

 

3.  Calculate the area and perimeter of the two “beds” in square kilometers.

Area in 2000                                                                Perimeter in 2000

 

Area in 2009                                                                Perimeter in 2009

 

 

4. What was the rate (in km2/ year) of expansion in the area of this population between 2000 and 2009?

 

 

 

 

 

5.  By what percent has the area occupied by the bed increased between 2000 and 2009?

 

 

 

6.  A third bed is discovered in 2009.  Again 1 cm represents 1 km.  What is the maximum length and width of this bed in km?

 

 

7.  Calculate the area of the oval bed, showing your work.

 

 

 

8.   Calculate the perimeter of the oval bed, showing your work.

 

 

 

9.  Which has the larger area?  The 2009 rectangular bed or the 2009 oval bed?

 

 

 

10.  Which has the greater ratio of perimeter : area?   The 2009 rectangular bed or the 2009 oval bed?

 

 

 

11.  Given that the beds spread from their perimeters, which bed would you expect to expand fastest (in terms of  % expansion) in the next few years? The 2009 rectangular bed or the 2009 oval bed?  Why?

 

 

 

 

 

 

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 http://www.zoomerang.com/Survey/?p=WEB229JA3BEWBD .  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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