Georgian Court University 

What speed the invasion? Estimating the rate of invasion of Phragmites in a Delaware MarshTime 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 reallife 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. · 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 Range, median, and mean o Calculators and computers used to record and process information 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:
Materials and Resources:
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.
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:
Easier: P = ( (3A+3B) – sqrt ((A+3B)*(B+3A)) ) Better but slightly more complicated is P = (a+b)(1 + 3x^{2}/[10 + sqrt(4  3x^{2})]) Where x = (ab) / (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:
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 (realworld scale). Closure Provide students with two yellow stickits. 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 stickits 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 ...
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 km^{2}/ 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?
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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.
