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The Annual Cycle of the Tree Farmer

Below is Lesson Plan #1 from The Annual Cycle of the Tree Farmer package, distributed free of charge by Massachusetts Christmas Tree Association.

The Annual Life Cycle of the Tree Farmer package is available for the cost of shipping only ($5.00). Please write or phone to place your order:

    C. Cranston
    197 Bellus Road / PO Box 77
    Ashland, MA 01330
    (413) 628-3911


Lesson 1 - How many trees can you grow?
Math, Science 1-8
High School - Environmental Science, Conservation Geometry

Suggestions for adapting the lesson for younger students are given at the end.

The following lesson is appropriate for grades 6 and up.

Objectives:
To develop an understanding of area,the size of an acre, and spatial relationships.
To learn different strategies for determining land area of irregular polygons

Problem:
Mr. and Mrs. Rice sold their dairy herd and want to convert the land that was formerly pasture into a tree farm. They have a map of their property that shows the pasture land. Literature from the Massachusetts Cooperative Extension agent about growing evergreen trees suggests that trees can be successfully grown in a six foot by six foot arrangement . Trees are planted six feet apart in rows that are six feet apart. With this arrangement 1200 trees can be planted per acre of pasture land. An acre is a surface area of 43,560 square feet.

Question to answer:
How many trees can the Rice Family plant in the former cow pasture?

A map of the Rice Property is included.

Teaching suggestions:
Day 1 Present the problem and do the following. Help students understand how large an acre is by taking them outside and measuring an acre of school property. Explain the following in the classroom. Since the square footage is 43,560 square feet, the easiest way to do this is to either create a square or a rectangle with this area.

A = L x W
Area of a square or rectangle = length x width
43,560 square feet(an acre) = length x width

Have students in the classroom using calculators experiment with numbers to find what combinations yield a product of 43,560.

Example 1
The square root of 43,560 is 208.47062 so one possibility is a square that is 208 ®.5 feet long and 208.5 feet wide 43,560 = 208.47 x 208.47

Other examples:
43,560 = 10 x 4356
43,560 = 100 x 435.6
43,560 = 150 x 43560 / 150
43,560 = 150 x 290.4


Students should discover that the second factor or width will always be equal to 43560 divided by the first factor, the length. (Math teachers can have students manipulate the equation mathematically by dividing both sides of the equation by the length to get the related equation : A/L = W. The class could then be divided in groups of four to select a specific combination that is an acre and to go outside to measure and stake out the bounds. This will help them understand that there are several different polygons that have the same area, an acre.

Day 2:
Finding areas of irregular polygons: the different pastures
Divide the class into groups of three or four. Hand out the maps of the farm that show the pasture land that is to be planted to trees. Have students focus on one of the pastures. Show this pasture on an overhead. Students are to spend five minutes discussing with members in their group how to find the square footage of this pasture. There are several possible solutions so let each group present how they found the area using the overhead.

Note: One of the keys to the solution is the scale of the map. Lengths on the map have to be converted to actual lengths using the map scale. If the map scale ratio is one inch represents 100 feet then 2.5 inches on the map is actually 250 feet. In this case one square inch on the map would be 100 feet by 100 feet and would represent 10,000 square feet or .229 acres.
(10,000 divided by 43560 = .229)

Possible ways to solve the problem:

    Find the sum of regular geometric shapes created in the interior. Divide the irregular shape into regular geometric shapes such as squares,rectangles, triangles, trapezoids. Find the area of these geometric shapes using the appropriate equation for area and total the results. It is best to create the largest geometric shape possible each time.

    Find the area of a regular geometric shape drawn outside the irregular shape and subtract the smaller areas not part of the pasture.

    Create a transparent grid of squares the sides of which equal the map scale. Place the grid over the pasture and count how many squares it takes to cover the pasture. Averaging may be used. If half or more of the square is within the bounds count is as a whole square. Less than half a square does not get counted. Multiply this number by the value of each square to get the acreage. The square value is determined by multiplying the real, not map length, of the side of a square by the length of the other side of the square and dividing this number by 43,560, the number of square feet in an acre. This will give the acreage represented by each grid square


Example.

    Assume each inch on the map represents an actual land distance of 200 feet.
    A grid of one inch squares is created.
    The actual length represented by each side of the square is 200 feet. The actual area represented by each square is 200 ft x 200ft or 40,000 square feet. Each square counted represents 40,000 divided by 43,560 or .918 acres.
    If 35 squares are counted to cover the pasture then the pasture is 35 squares x .918 acre per square = 32.13 acres in area.
    Since 1200 trees can be planted per acre this pasture can be planted to: 32.13 acres x 1200 trees per acre, or 38,556 trees.


Find the acreage for each pasture and the number of trees that can be planted in each of the pastures.

Note for younger students: Instead of using the map included in the packet create your own map with easier shapes to work with. Perhaps the pasture shape is a trapezoid. A map of one of the pastures where this has already been done is included. Students could get the area by dividing the trapezoid into a rectangle and one or more triangles. Perhaps the pasture is a series of different shaped rectangles. Use whatever is appropriate for your students level of understanding. Feel free to create your own version of the pastures.

For very young students: Students are to pretend that they are trees. Each tree needs to have enough space to grow When trees do not have enough space their branches touch and brush against each other. The needles are broken and sometimes the branches break off. To grow optimally the trees need about three feet of space in every direction. Students can find the tree space boundaries by spreading their arms out horizontally in every direction until they just touch another person's (tree) fingertips. Extend the lesson by having the students determine how many trees could be planted in an area of the playground. (Perhaps a baseball diamond.)

Math Practice for grades 3-6: (Each acre will grow 1200 trees.)

Pasture

Total acres

Roadways

Acres growing trees

# of trees

1

5.5

5

 

 

2

15

2.5

 

 

3

3.0

.5

 

 

4

4

1.0

 

 

5

4.5

.5

 

 

* More problems can be added for additional practice. Students are to find how many acres are growing trees and the total number of trees growing in each pasture.

 

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