Forestry - EE In Wisconsineeinwisconsin.org/Files/eewi/2011/Unit06Forestry.pdf · Forestry About...

Forestry

About this UnitThe purpose of this unit is to answer the question of how many trees do each of us in the United States use annually. A variety of math skills will be used to answer this question. Math topics that are covered in this unit include geometry (use of similar and right triangles), volume calculations, area calculations, sampling, graphing, coordinate systems, basic probability, and scale.

Students will utilize similar triangles to make a tree scale stick that will be used to measure the height and diameter of trees. They will also use right triangles to estimate the height of the tree and circle geometry to make a diameter tape to compare these estimates with those from their tree scale stick estimates. Students will then establish a plot to sample the species, diameters, and heights of trees in a forest. A coordinate system will be used to locate each of the trees in a plot. The data collected from the plot will then be used to calculate volumes of the trees in the plot and relate that to a per area (acre) basis. Trees will be graphed by species, volume, and diameter. Mean, mode, median, range, and probability will be calculated. Students will make a scale drawing of their team plots and the drawings will be combined to make a mural of their plots. To conclude the unit, students will present the data from their plots to answer the question of how many trees do each of us use annually. A discussion will be held to get students’ perspectives on their consumption.

Contributing Writer

Forestry | | 1 |

Jeremy Solin

Jeremy is an environmental educator whose focus is community sustainability. He has a background in forestry and natural resource management. He’s worked in the non-formal education field for approximately 10 years in positions in Minnesota, Oregon, and Wisconsin. He’s

currently the director of and was previously the statewide school forest coordinator for the LEAF K-12 Forestry Education Program at UW-Stevens Point. He has a bachelor’s degree in water resources and master’s degree in education.

| 2 | Forestry | Unit 6

Forestry

Lessons included in this unit:

Lesson 1: Unit introduction: How do we use wood? . . . . . 5Skill Building: How much is 75 cubic feet?

Lesson 2: Tree Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Skill Building: Triangles Angles and Sides

Skill Building: Similar Triangles

Skill Building: Using Similar Triangles

Skill Building: Similar Triangle Search

Skill Building: Right Triangles

Skill Building: Using Isosceles Right Triangles

Final Project: Tree height measurements

Lesson 3: Tree Diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Skill Building: Circle Geometry

Final Project: Tree diameter measurements

Lesson 4: Tree Plot Study . . . . . . . . . . . . . . . . . . . . . . . . . . 27Skill Building: Coordinate System

Final Project: Plot Study

Lesson 5: Using the Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Skill Building: Tree Volume

Skill Building: Statistics and Probability

Final Project: Plot Sample Summary

Lesson 6: Presentation: How Many Trees? . . . . . . . . . . . . 42Final Project: Project Presentation

| 3 |Forestry | Unit 6

Notes for unit planning:

Forestry• is organized around the driving question of how many trees are required to meet our annual forest product consumption. There are six lessons containing multiple activities. The lessons are designed in a sequential format, although it is possible to just complete some of the activities. However, to answer the driving question and complete the final projects (map and presentation), each of the lessons needs to be completed.

The entire unit will require approximately 20-25 hours to complete. Aside from the introductory lesson, which is •just an hour long, each lesson is approximately 3-5 hours in length.

The first three lessons provide the background information, geometry, and tools to complete the rest of the unit. •Depending on the students, more extensive activities in similar and right triangles may be necessary for students to fully understand the concepts. See the resource section for sources of more activities.

Each lesson contains assessment activities which are integrated into the lesson. The final projects provide an •application of and assessment for the skills and knowledge gained throughout the unit.

Each lesson contains worksheets and data sheets for student use. However, a notebook or other recording •mechanism (laptop computer) can be used in place of the sheets.

Content:There is considerable use of similar and right isosceles triangle and circle geometry in the unit. Basic statistics and probability are used to analyze the results of a plot study data collection. Coordinate systems are used to locate and map trees in a plot. Ratios and scaling are addressed through a mapping activity. Students create graphs in analyzing data. The type and extent of the graphing are optional and will depend on the goal for the lesson and previous experience with graphing. A compass is used to establish coordinates in the plot study. A link to teaching compass basics is provided, but it would be helpful if students are proficient with a compass.

Non-math concepts:The primary non-math environmental concepts are forestry and resource consumption. The entire unit deals with the consumption of forest products and students are asked to consider their perspective relative to the degree of our consumption. The forestry concepts are basic, primarily dealing with forest measurements, but provide a starting point for further investigation if desired.

Additional possibilities:There are host of additional and alternative possibilities within and following the unit. Instead of forest product consumption, the driving question could deal with carbon sequestration and the amount of forest land needed to store the carbon each person in the United States is responsible for. The unit could also take a more ecological approach and look just at productivity of the forest—the amount of annual growth of the trees.

The exploration of how many trees could be extended by examining the age of the forest through various means— input from a forester, boring samples trees to determine their age, interviewing neighbors familiar with the history of the land, etc. This would then provide not only an area basis for how many trees, but also a length of time.

Looking at the management of the forest to examine management methods, management plan development, and local forest product markets would be a good follow up to this unit.

The unit could also include a detailed inventory of forest product utilization and attempts to calculate the volume of forest products consumed. A comparison between the environmental impacts of forest products and other non-renewable resources could also be conducted.

| 4 | Forestry | Unit 6

Forestry

Unit Vocabulary

acre – an area of land equal to 43,560 square feet.

calibrate – to check, adjust, or determine by comparison with a standard.

chain – a length equal to 66 feet. This is a common forestry measurement.

consumption – in this unit, it has to do with how much of a resource is used.

dbh – diameter at breast height is the standard diameter measurement of a tree. It is measured 4.5 feet above the ground and recorded in inches.

even-aged forest – a forest where most of the trees are approximately the same age and near the same height and diameter.

experimental probability – the ratio of the number of times the event occurs to the total number of trials P(E) = number of successful outcomes ÷ total number of outcomes.

frustum – the part of a solid, as a cone or pyramid, between two usually parallel cutting planes.

hypsometer – a tool to measure height of trees.

isosceles right triangle – a triangle with a 90º angle and two equal sides.

mean – what is usually meant by the “average,” computed by adding all pieces of data and then dividing by the total number of pieces (also known as the arithmetic mean).

median – the middle number.

merchantable height – the amount of tree that can be used for forest products. For this lesson, merchantable height is from 6" above the ground to the point where the diameter of the tree reaches 4".

mode – the number that occurs the most often.

per capita – per person.

pi – The ratio of the circumference of a circle to its diameter, π ≈ 3.14159 26535 89793...

plot – is a sample area for the purpose of collected representative data.

proportional – two quantities with a constant ratio.

range – in statistics, the difference between the largest and the smallest numbers in a data set.

sampling – gathering data from a small population to represent information about an entire group.

similar triangles – triangles that have the same shape but not necessarily the same size; correspond-ing sides are in proportion and corresponding angles are congruent.

tree caliper – device used to measure the diameter of a tree.

tree diameter – the diameter of the cross-section of a tree at a given height. For this purpose of this unit, tree diameter will be measured at 6” above the ground.

tree height – can be measured for many different purposes. Total tree height and merchantable tree height can vary considerably.

uneven-aged forest – a forest where the trees vary in age and size.

Forestry | Lesson 1 | 5 |

Forestry

Lesson Plan How do we use wood? • Lesson 1

Learning ObjectivesStudents will be able to list 10 1. things that they use or benefit from on a daily basis that are made or come from trees.Students will be able to 2. represent the volume of wood products they use each year.Students will be inspired to explore 3. how many trees it takes to meet their annual use of wood.

Materials12 inch rulers ❒

Student worksheet (one per student) ❒

PreparationResearch common products made 1. from wood. For a good overview of forest products, read the ar-ticle “What’s a Tree Done For You Lately?” available at: http://owic.oregonstate.edu/teachers.php.

Hour 1(15 min) Introduce the unit. Provide an introduction and

overview of the “How Many Trees” unit. Explain the math concepts and skills that will be utilized. Share the activities and projects that will be conducted during the course of the unit.

(20 min) Brainstorming forest products. Ask students to think of a tree product that they have used today. Take one or two answers as examples. Explain to students that we all use or benefit from tree products on a daily basis. Have students take out a notebook and record as many tree products as they can think of. Remind students to think of products at home, at school, things they use for recreation, etc. If you think it will be helpful, you can introduce categories of forest products such as fiber (paper-like products), solid wood products, food products, and chemical products. If computers are available, students can also do an internet search for wood products that they use.

Review the product lists that students have generated. Record these as a master list visible to all students. If needed, add to the list. To see a list of things of things that are and used to be made of wood, visit this site: http://www.uky.edu/Ag/Forestry/conners/WoodUses.pdf

(5 min) How much wood do we all use? Ask the students if they have any idea how much wood they use every year. Have them make a hypothesis. The average annual consumption of each United States citizen is about 75 ft3 (about 1 ton depending on how dry the wood is). This includes all of the paper and fiber products, solid wood products, wood chemical, and some of the food products from wood that we use (food products such as apples are not included in this).

( 20 min) How much wood is that? Students calculate the number of common things that equal our annual volume utilization of wood

Using the worksheet, have students convert 75 ft3into more familiar terms. Review the worksheet with the students.

Review the answers with the students. Explain that all of the rest of the activities in this

unit will be focused on answering the question of how many trees does it take to make the 75 ft3 of wood that they use each year.

Ask them, were their hypothesis correct? Close? Way off? Were their answers surprising?

http://owic.oregonstate.edu/teachers.php

http://owic.oregonstate.edu/teachers.php

http://www.uky.edu/Ag/Forestry/conners/WoodUses.pdf

http://www.uky.edu/Ag/Forestry/conners/WoodUses.pdf

| 6 || 6 | Forestry | Lesson 1

Name ��Skill Building

How Much is 75 Cubic Feet?Remember:Showyourworkandmakesureyouusethecorrectlabelonallyouranswers!Youwillneedacalculatorforthisexercise.Roundtothenearest100th.

Each of us in the United States uses about 75 cubic feet of wood products. This includes things like wood in our houses and other buildings, paper products, food additives (cellulose), medicines, and cleaning products.Directions: In this activity, you will get a better idea of how much 75 cubic feet is. Calculate the volumes of different every day things to see how many of those things are needed to equal 75 cubic feet. To calculate the volume of a solid object, you need to know its length, width, and height in feet. For fluid materials (like soda), you’ll use a known volume.

Object Measurements Volume (cubic feet)

Number to equal 75 cubic feet

Length(in feet)

Width(in feet)

Height(in feet)

Length x Width x Height

Divide 75 by the volume

Liters of soda - - - .035 cubic feet

Sheets of paper 0.92 feet (11 inches)

0.7083 feet (8.5 inches) .000344 feet (round to nearest

millionth here)

Cell phone

Textbook

Locker

Gallon of milk - - - .1337 cubic feet

Showworkhere:


Forestry

Lesson Plan Tree Height • Lesson 2

Learning ObjectivesStudents will be able to utilize 1. similar triangle geometry to measure the height of a tree.Students will be able to use 2. right triangle geometry to measure the height of a tree.Students will be able to draw 3. conclusion about the accuracy and variability of two different methods to estimate tree height.

MaterialsProtractors (one per student) ❒

12" Rulers (one per student) ❒

Flat, smooth sticks/boards or ❒other rigid material about 1 inch wide, less than and inch thick, and at least 48 inches long – lathe works well (one per student)100' tape measures or 100' long ropes ❒with markings indicating each foot (one per pair of students) (alterna-tively students can learn pacing and use their pace to measure distances)Yard stick (one per ❒student or pair of students)Permanent markers ❒

Worksheets (one of each ❒worksheet per student)

PreparationBe comfortable with similar ❒and right trianglesLocate a place and make ❒arrangements for students to visit a location with trees for students to measure. This can be a schoolyard or park with at least 10 trees.Label the trees 1-10. ❒

Hour 1Warm up: In this lesson we will learn that the inside angles always

add up to 180º. Is it magic? Why might that be the case? Look at the sum of the angles of a square, pentagon and a hexagon. Are they 180º degrees as well? Why not? Lead the students to see that there is a formula associated with thesumoftheinneranglesofashape:(#sides−2)•180º

(30 min) Introduce triangles(30 min) Introduce similar triangles

Hour 2(10 min) Similar triangle review and practice(20 min) Introduce right triangles(30 min) Right triangle practice

Hour 3(60 min) Using similar and right triangles to make a tree height stick

Hour 4(60 min) Measuring trees

Using the tree height dataDiscussion: Which types of trees were the tallest?

Were the trees mostly consumable or were they filled with lots of small branches?

Lesson Extension: To review what volume really means, have students

construct a tree out of PVC pipes (or other plas-tic pipes). Packaging tape could be used to tape on a couple of branches. Pour cups of water and count how many it takes to fill. 1 ft3 ≈ 119.7 cups

1. IntroductionProvide an overview of this lesson by letting students •know that now that they know how much wood each of uses, they are going to begin gaining the skills to help them determine how many trees are needed to equal 75 cubic feet. Ask students what measurements will be needed to calculate a tree’s volume. (They will need diameter – diameter is used to calculate the surface area of the tree - and height). Discuss with students that these measurements are difficult to get directly because of the heights of the trees and the curvature of the tree trunks. They will be using geometry to make estimates of the tree beginning with height and then going to diameter.

Forestry | Lesson 2| 8 |

2. Introduce triangles Ask students what the characteristics of triangles are (3 sided shape with 3 angles). Explain that •triangles can be described by the lengths of their sides and their angles. Using the “Triangle Angles and Sides” worksheet provided, have the students measure the angles and sides of the triangles and calculate the sum of those angles. Point out that the sum of all the angles in a triangle is 180 degrees.Introduce similar triangles• . Ask students what it means to be similar (to have many things in common, but not everything is the same). When we say that objects are similar, it means they have the same characteristics, but not all the same measurements. This can be thought of as a difference in scale—that is, if an object is enlarged or shrunk proportionally (all aspects equally), it will be similar to the original. Show students the visuals of similar objects using the “Similar Objects” transparency. Similar triangles are triangles that have the same three angles. However, since the sum of the angles in a triangle is always 180 degrees, as long as two angles are the same, all three are. If the angles are the same, the sides are proportional. Using the “Similar Triangles” worksheet, have students identify the characteristics of similar •triangles. Have the students measure the sides and angles of the pairs of triangles given. They should find the respective angles of the triangles the same. The ratio of each respective side should also be the same. To calculate the ratio, have students use a calculator to divide the lengths of triangle 1 sides by the corresponding side of triangle 2. Help students with the calculations if necessary. Review the answers with the class. The answers to the division problems should be the same.Show students how the properties of similar triangles can be used to calculate the side •lengths of a triangle given one side measurement and the measurements of the similar triangle using ratios. Have students complete the “Using Similar Triangles” worksheet and review the answers. If students have not done cross multiplication, you will likely need to provide additional instruction for the students to feel comfortable accomplishing this.Review similar triangles. • Using the “Similar Triangle Search” worksheet, have students match similar triangles. Review answers with students. (Similar triangles are: 1-10, 2-3, 4-5, 6-9, 7-11, 8-12) Introduce right triangles. • Right triangles are triangles with one 90 degree angle. The 90 degree angle is called a right angle. Ask students if they have worked with any right triangles during the last couple activities. Help students select which triangles were right triangles, if necessary. Ask students again what the sum of all the angles in a triangle is (180). If all the angles add up to 180 degrees, what is the sum of the other two angles in a right triangle? Explain to students that right triangles have some unique properties that will be used to help them measure trees. The triangle that the students will find most useful is the isosceles right triangle. This triangle has 2 sides of equal length and two 45º angles. Have the students explore the right triangles using the “Right Triangles” and “Using Right Isosceles Triangles” worksheets provided.

3. Make tree height sticks. Using similar and isosceles right triangles, students will make sticks that they will use to estimate •the height of trees. Each stick will be unique to each student as the sticks will be based on each student’s arm reach. One side of the piece of wood will be used for the diameter scale and the other will be for tree height. Begin by having students measure their arm reach using a yard stick. Working in pairs have one students of each pair completely extend their arm holding the “0” end of the yard stick and extending the yard stick vertically. With the student keeping their arm extended, the other student should tilt the yard stick and measure the distance from the fingers to the eye of student with his/her arm extended. Make sure each student records this number.

http://en.wikipedia.org/wiki/Angle


The height sticks will have two sets of markings: one of tree height calibrations •using similar triangles and a single mark representing a 45º angle.

For the similar triangle calibrations:The height stick (or hypsometer) is made using the similar triangle calculations. Show students •and explain the similar triangles that they will be using to determine the tree heights. The students’ eyes will form one point of a triangle. The points of eyesight on the stick will form the other two points of the first triangle. The second triangle is formed by the students’ eyes and the top and bottom of the tree. The one difference with these similar triangles and what students have already seen is that the two similar triangles are made up of 2 pairs of similar right triangles. Because of the geometry of these pairs of similar triangles, they can be combined to form the larger similar triangles while using the characteristics of those right triangles.

As a result, •

€

DEAG

=

€

BCAF .

The tree scale stick-eye triangle and the tree-eye triangle are similar triangles, as all of the angles are the same.

To create the calibrations on the tree diameter stick, you need to work backward from the •tree height to determine where each height mark is on the stick (this is the length BC). The challenging part of this calculation is the conversion between inches and feet. It will be most •accurate to convert feet to inches to determine the calibrations on the stick. The formula is

€

DEAG

=

€

BCAF

, AG and AF are known measurements. In addition to the known arm length, a set distance from the tree needs to be determined. Typically this distance is set at 66 feet (66 feet equals one chain, which is a standard forestry and surveying distance).

DE (height of the tree) can be done in increments of 5 or 10 feet beginning at 10 feet and going up •to 100 feet. For example, to determine the calibration for a 20 foot tall tree and a 25 inch arm reach:

Tree Height=

Calibration MarkDistance to Tree Arm Reach

Triangle ABC is the triangle •formed by the eye and the stick. Triangle ADE is the triangle •formed by the eye and the top and bottom of the tree.They are similar triangles •because the angles are the same.The length of AF is the •arm reach.The length of AG is the •distance from the person to the tree. The length BC is the •calibration mark.The length DE is the •height of the tree.

Cross multiply and divide: BC = 240" × 25"÷792" = 7.6"

| 10 | Forestry | Lesson 2

Students can be given the similar triangles and can calculate the calibrations on the stick. •However, if after showing students how the calculations are done you just want them to skip to making the sticks, they can use the chart “Tree Height Calibrations” provided.

For Using Isosceles Right Triangle Method: Ask student about the properties •of isosceles right triangles. Ask them how these properties can be used to determine the height of the tree (if you can determine a 45º angle to the top of the tree, the height of the tree is equal to the distance from the tree as the sides of an isosceles triangle are equal). Ask students how they can determine •if a 45º angle is formed to the top of a tree. (Answers could include a protractor or making a mark on their tree height stick that represents the “sight line” for a 45º angle). Ask if any students can determine an easy way, using the properties of isosceles right triangles to determine where this mark goes (a distance from the bottom end of the stick equal to their arm reach).

NOTE: the 45º angle is formed from the student’s eye to the top of the tree. To be accurate in determining the height of the tree, the distance from the ground to the eye level should be added to height determined by distance from the tree.

4. Measuring trees. Take students to a nearby area to measure trees. Locate and label 10 trees at least 10 •feet tall. Have students determine the height of each of the trees using both similar triangle and isosceles right triangle calibrations on their tree height stick. Students can record their data on the “Tree Height Measurements” data sheet. Using the tree height data. Students should compare the estimates they got for the tree heights. •Individually have students compare their two different methods for tree height. Was there a consistent difference (did one method consistently provide a higher height estimate)? If so, why might that be?Using all the estimates made by the class for the ten trees, have students calculate the •average and range of the data. Can conclusions be drawn about the most accurate method to estimate tree height? Was there a wide range of estimates for each tree? If so, why?Conclusion of tree heights: remind students that they now have the skills and knowledge •to obtain one aspect of the information they need to determine how many trees they use each year. Next they will be working to find out the diameter of the trees.



Triangle Angles and Sides

1. Angle A = �� degrees

2. Angle B = �� degrees

3. Angle C = �� degrees

4. Sum of angles = �� degrees

5. Side AB = �� inches

6. Side BC = �� inches

7. Side CA = �� inches
















Name ��Transparency

Similar Objects

What makes these objects similar? What do you notice about the angles formed by these pictures?

Non-Similar Objects



Triangle 1






6. Side AC = �� inches

Triangle 2

7. Angle a = �� degrees

8. Angle b = �� degrees

9. Angle c = �� degrees

10. Side ab = �� inches

11. Side bc = �� inches

12. Side ac = �� inches

Triangle 1 vs Triangle 2 Ratio of sides

13. Tri 1 AB : Tri 2 ab = ��

14. Tri 1 BC : Tri 2 bc = ��

15. Tri 1 AC : Tri 2 ac = ��

Similar TrianglesDirections: Measure the angles and sides of the similar triangles

Triangle 1

a

b c

Triangle 2



Triangle 3






21. Side AC = �� inches

Triangle 4

22. Angle a = �� degrees

23. Angle b = �� degrees

24. Angle c = �� degrees

25. Side ab = �� inches

26. Side bc = �� inches

27. Side ac = �� inches

Triangle 3 vs Triangle 4Ratio of sides

28. Tri 3 AB : Tri 4 ab = ��

29. Tri 3 BC : Tri 4 bc = ��

30. Tri 3 AC : Tri 4 ac = ��

Triangle 3

a

bc

Triangle 4

A

CB

Similar Triangles, page 2Directions: Measure the angles and sides of the similar triangles



Using Similar TrianglesWhen triangles are similar, their angles are the same and their measurements are proportional. These properties can be used to determine the unknown lengths of one triangle’s sides if the lengths of the other triangle’s sides are known.

Directions: Given the length of the sides for two of the triangles and the fact that each set of the triangles are similar, calculate the side lengths for triangle 2 and 4. Show your work and round to the nearest 100th.ound to the nearest 100th.

Similar Triangle Calculations

Triangle1andTriangle2aresimilar

1. Triangle 1Side AB = 12 inchesSide BC = 10 inchesSide AC = 15.6 inches

2. Triangle 2Side AB = �� inchesSide BC = �� inchesSide AC = 10 inches

Triangle3andTriangle4aresimilar

3. Triangle 3Side AB = 4 inchesSide BC = 5 inchesSide AC = 3.2 inches

4. Triangle 4Side EC = �� inchesSide DE = 1.9 inchesSide DC = �� inches

Triangle Triangle



Similar Triangle SearchDirections: Match the similar triangles – You may need to measure the sides or angles to find similar triangles.



Right TrianglesRemember:Showyourworkandmakesureyouusethecorrectlabelonallyouranswers!

Directions: Measure the angles and sides of the right triangles

Triangle 1

Angle A = �� degrees Side AB = �� inches

Angle B = �� degrees Side BC = �� inches

Angle C = �� degrees Side AC = �� inches

Triangle 2




Triangle 3




Triangle 4




Which triangle is the isosceles right triangle?



Using Isosceles Right TrianglesDirections: Complete the angles and side lengths using the properties of right isosceles triangles (no need to measure) Round to the nearest 10th.

Isosceles Right Triangle 1

Angle A = �� degrees

Angle B = �� degrees

Angle C = �� degrees

If side AB is 77 feet long, how long is side BC? Remember the Pythagorean theorem!

a2 + b2 = c2

Triangle 2The angle from the person to the top of the tree is 45 degrees.

The tree is perfectly perpendicular to the ground (its trunk makes a 90 degree angle with the ground)

The person is standing 41 feet from the tree.

How tall is the tree?



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3.64

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4.55

4.77

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5.23

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6.59

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204.

554.

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766.

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366.

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925

.45

26.5

227

.58

28.6

429

.70

30.7

631

.82

7517

.05

18.1

819

.32

20.4

521

.59

22.7

323

.86

25.0

026

.14

27.2

728

.41

29.5

530

.68

31.8

232

.95

34.0

980

18.1

819

.39

20.6

121

.82

23.0

324

.24

25.4

526

.67

27.8

829

.09

30.3

031

.52

32.7

333

.94

35.1

536

.36

8519

.32

20.6

121

.89

23.1

824

.47

25.7

627

.05

28.3

329

.62

30.9

132

.20

33.4

834

.77

36.0

637

.35

38.6

490

20.4

521

.82

23.1

824

.55

25.9

127

.27

28.6

430

.00

31.3

632

.73

34.0

935

.45

36.8

238

.18

39.5

540

.91

9521

.59

23.0

324

.47

25.9

127

.35

28.7

930

.23

31.6

733

.11

34.5

535

.98

37.4

238

.86

40.3

041

.74

43.1

810

022

.73

24.2

425

.76

27.2

728

.79

30.3

031

.82

33.3

334

.85

36.3

637

.88

39.3

940

.91

42.4

243

.94

45.4

5

Tree Height CalibrationHeightCalibrationforTreeStickwithDifferentArmReaches



Tree Height MeasurementsRemember:Showyourworkandmakesureyouusethecorrectlabelonallyouranswers!

Directions: Complete the table below:

Tree Number

Height Using Similar Triangle Calibrations Height Using Right Triangle Average Tree Height

(of both methods)

1

2

3

4

5

6

7

8

9

10


Forestry

Lesson Plan Tree Diameter • Lesson 3

Learning ObjectivesStudents will be able to utilize 1. similar triangle geometry to mea-sure the diameter of a tree.Students will be able to use 2. circle geometry to measure the diameter of a tree.Students will be able to draw 3. conclusion about the accuracy and variability of the two methods to determine a tree’s diameter.

Materials12" Rulers (one per student) ❒

Tree height sticks – one side will ❒be used for a tree diameter stick100" lengths of rope or ❒ribbon (one per student)Yard stick (one per student ❒or pair of students)Permanent markers ❒

Worksheets (one of each ❒worksheet per student)

PreparationBe comfortable with similar ❒triangles and circle geometry.Plan to visit the site from lesson 2 to ❒find trees for students to measure.

Hour 1(10 min) Introduce tree diameter(50 min) Introduce and practice with circle geometry

Hour 2(5 min) Review circle geometry(20 min) Using circle geometry to make a diameter tape(35 min) Using similar triangles to make a tree diameter stick During this hour, it might be helpful for students to

practice their measurements on tubes and cylinders.

Hour 3(30 min) Measuring trees(30 min) Using the tree diameter data

(5 min) Conclusion

1. Introduce diameter Ask students what measurements will be needed to calculate •the volume of trees (diameter and height). Ask students how that will done (how do you get a linear measurement from a circular object)? There are a number of possibilities: cutting the tree, tree caliper, measure the circumference, use similar triangles, etc. Tell the students they are going to learn two ways to estimate a tree’s diameter. Again, this will require some geometry – some they know and some new.Tree diameter: Obviously the diameter of a tree depends •on where it’s measured. The diameter at the top is much different than at the bottom. Foresters typically measure the diameter of the tree at 4 ½ feet above the ground. This is called dbh – diameter at breast height. Diameter is measured here because it is easy – easier than bending down to ground level each time. However, because the students are interested in knowing the total volume of the tree, they are going to measure the tree diameter at stump level (about 6" above the ground).

2. The geometry of tree diameters Trees are assumed to be approximately a circle in cross •section. Because of this, the geometry of a circle can be used as one way to estimate the diameter of the tree. Share with and explain to students the formulas relating circles’ radius, diameter, circumference, and area.


C = πd 1. A = πr2. 2

Have students work with these formulas using the “Circle Geometry” worksheet. Review the •answers with the students. Remind the students that, along with tree height, the tree surface area (area of the circular cross section of the tree) is used to calculate tree volume.

3. Making a tree diameter tape.Have students make a tool to measure the diameter of trees from one inch to 30 inches. •A mark/gradation should be made for each inch of diameter. Provide students with rope, ribbon, or cord, a yard stick, and markers to make their diameter tape using the formula for the circumference of a circle. (Some students will likely catch on quickly that each gradation is 3.14 inches apart and won’t need to do the calculations).

4. Making the diameter stick. Students will add diameter calibration to the other side of their tree height •stick. Again, students will need to know their arm reach and will be using similar triangles to create the diameter stick calibrations.Show students and explain the similar triangles that they will be using to make the stick to determine •the diameter of trees. The students’ eyes will form one point of a triangle. The points of eyesight on the stick will form the other two points of the first triangle. The second triangle is formed by the students’ eyes and the edges of the tree. The tree scale stick eye triangle and the tree sides eye triangle are similar triangles as all of the angles are the same.

Remind students that the similar triangles are made up of 2 pairs of similar right triangles. •Because of the geometry of these pairs of similar triangles, they can be combined to form the larger similar triangles while using the characteristics of those right triangles. As a result, DE/AG = BC/AF. This allows us to create a tree diameter stick, with one additional assumption. That assumption is that the tree cross-section is a circle. If the tree cross section

Triangle ABC is the •triangle formed by the eye and the stick.Triangle ADE is the •triangle formed by the eye and the edges of the tree.They are similar triangles •because the angles are the same.The length of AF is the •arm reach.The length BC is the •calibration mark.The length AG is the arm •reach plus the radius of the tree.The length of DE is the •diameter of the tree.


is a circle, then AG = AF + the radius of the tree. AF is a known length (arm reach). With their known arm reaches, the formula becomes BC (the calibration •mark) = tree diameter x arm reach / (tree radius + arm reach).To create the calibrations on the tree diameter stick, you need to work backward from the •tree diameter to determine where each diameter marking is on the stick (this is the length BC). For example, for a 25" arm reach (AF) and a 6" diameter tree, the calculation is:

Variables: AF = 25 AG = 25 + 3 (radius of a 6" tree) = 28 DE = 6 BC = ?

Formula:DE

=BC

AG AF

Therefore, a mark representing a 6" diameter tree would be made 5.35 inches from the end of the stick. •Have students begin with a mark for 4" in diameter and continue up to 30" (unless you •know only smaller trees will be measured). Increments of 1" or 2" can be made.The students can use similar triangles to calculate the calibrations on the stick. Show the •students how the calculations are done. If the calculations are too difficult or too time consuming, skip the calculations and use the “Tree Diameter Calibrations” table provided.

5. Measuring trees. Take students to the area used in lesson 2. Locate and label the 10 trees used previously, making •sure that they are at least 4" in diameter (because of the frustum). Remember that the trees will be measured at stump level, about 6" above the ground. Show students how to measure the diameters of the trees using the diameter stick and the diameter tape. Hold the diameter stick at arm length (parallel to the ground) and align one edge of the stick with one edge of the trunk and sight to where the other edge of the trunk crosses the stick. The diameter tape is simply wrapped around the tree at the desired height and the diameter is read where the tape meets the zero end of the tape. Have students record the diameters of each of the trees using both the tree diameter stick and diameter tape using the “Tree Diameters” data sheet.

6. Using the tree diameter data. Students should compare the estimates they got for the tree diameters. Individually have students •compare their two different methods for tree diameter. Was there a consistent difference (did one method consistently provide a higher diameter estimate)? If so, why might that be?Using all of the estimates made by the class for the ten trees, have students calculate the •average and range of the data. Can any conclusions be drawn about the most accurate method to estimate tree diameter? Was there a wide range of estimates for each tree? If so, why? (Note: Typically, the tree diameter tape will provide the most accurate estimate as it is more of a direct measurement and also incorporates variation in tree shape.)

7. Conclusion of tree heights: Remind students that they now have the skills and knowledge to obtain both aspects of •the information they need to determine how many trees they use each year – the height and diameter. Next they will conduct a study to make the volume calculations.

; 150 = 28x; x = 5.35



Circle GeometryCircumference of a circle = pi • diameter

Area of circle = pi • radius2

Circle One

Diameter = �� cm

Radius = �� cm

Circumference = �� cm

Area = �� cm2

Circle Two


Radius = �� cm


Area = �� cm2

Circle Three


Radius = 25.4 cm


Area = �� cm2

Circle Four


Radius = �� cm

Circumference = 55.9 cm

Area = �� cm2

Circle 1

Circle 2

Given the measurement, determine the following:

Remembertoshowyourworkandroundtothenearest10th.



Dia

met

er C

alib

ratio

ns fo

r Tre

e St

ick

with

Diff

eren

t Arm

Rea

ches

Arm

Rea

chTr

ee

Dia

met

er

(inch

es)

1516

1718

1920

2122

2324

2526

2728

2930

43.

533.

563.

583.

603.

623.

643.

653.

673.

683.

693.

703.

713.

723.

733.

743.

755

4.29

4.32

4.36

4.39

4.42

4.44

4.47

4.49

4.51

4.53

4.55

4.56

4.58

4.59

4.60

4.62

65.

005.

055.

105.

145.

185.

225.

255.

285.

315.

335.

365.

385.

405.

425.

445.

457

5.68

5.74

5.80

5.86

5.91

5.96

6.00

6.04

6.08

6.11

6.14

6.17

6.20

6.22

6.25

6.27

86.

326.

406.

486.

556.

616.

676.

726.

776.

816.

866.

906.

936.

977.

007.

037.

069

6.92

7.02

7.12

7.20

7.28

7.35

7.41

7.47

7.53

7.58

7.63

7.67

7.71

7.75

7.79

7.83

107.

507.

627.

737.

837.

928.

008.

088.

158.

218.

288.

338.

398.

448.

488.

538.

5711

8.05

8.19

8.31

8.43

8.53

8.63

8.72

8.80

8.88

8.95

9.02

9.08

9.14

9.19

9.25

9.30

128.

578.

738.

879.

009.

129.

239.

339.

439.

529.

609.

689.

759.

829.

889.

9410

.00

139.

079.

249.

409.

559.

699.

819.

9310

.04

10.1

410

.23

10.3

210

.40

10.4

810

.55

10.6

210

.68

149.

559.

749.

9210

.08

10.2

310

.37

10.5

010

.62

10.7

310

.84

10.9

411

.03

11.1

211

.20

11.2

811

.35

1510

.00

10.2

110

.41

10.5

910

.75

10.9

111

.05

11.1

911

.31

11.4

311

.54

11.6

411

.74

11.8

311

.92

12.0

016

10.4

310

.67

10.8

811

.08

11.2

611

.43

11.5

911

.73

11.8

712

.00

12.1

212

.24

12.3

412

.44

12.5

412

.63

1710

.85

11.1

011

.33

11.5

511

.75

11.9

312

.10

12.2

612

.41

12.5

512

.69

12.8

112

.93

13.0

413

.15

13.2

518

11.2

511

.52

11.7

712

.00

12.2

112

.41

12.6

012

.77

12.9

413

.09

13.2

413

.37

13.5

013

.62

13.7

413

.85

1911

.63

11.9

212

.19

12.4

412

.67

12.8

813

.08

13.2

713

.45

13.6

113

.77

13.9

214

.05

14.1

914

.31

14.4

320

12.0

012

.31

12.5

912

.86

13.1

013

.33

13.5

513

.75

13.9

414

.12

14.2

914

.44

14.5

914

.74

14.8

715

.00

2112

.35

12.6

812

.98

13.2

613

.53

13.7

714

.00

14.2

214

.42

14.6

114

.79

14.9

615

.12

15.2

715

.42

15.5

622

12.6

913

.04

13.3

613

.66

13.9

314

.19

14.4

414

.67

14.8

815

.09

15.2

815

.46

15.6

315

.79

15.9

516

.10

2313

.02

13.3

813

.72

14.0

314

.33

14.6

014

.86

15.1

015

.33

15.5

515

.75

15.9

516

.13

16.3

016

.47

16.6

324

13.3

313

.71

14.0

714

.40

14.7

115

.00

15.2

715

.53

15.7

716

.00

16.2

216

.42

16.6

216

.80

16.9

817

.14

2513

.64

14.0

414

.41

14.7

515

.08

15.3

815

.67

15.9

416

.20

16.4

416

.67

16.8

817

.09

17.2

817

.47

17.6

526

13.9

314

.34

14.7

315

.10

15.4

415

.76

16.0

616

.34

16.6

116

.86

17.1

117

.33

17.5

517

.76

17.9

518

.14

2714

.21

14.6

415

.05

15.4

315

.78

16.1

216

.43

16.7

317

.01

17.2

817

.53

17.7

718

.00

18.2

218

.42

18.6

228

14.4

814

.93

15.3

515

.75

16.1

216

.47

16.8

017

.11

17.4

117

.68

17.9

518

.20

18.4

418

.67

18.8

819

.09

2914

.75

15.2

115

.65

16.0

616

.45

16.8

117

.15

17.4

817

.79

18.0

818

.35

18.6

218

.87

19.1

119

.33

19.5

530

15.0

015

.48

15.9

416

.36

16.7

617

.14

17.5

017

.84

18.1

618

.46

18.7

519

.02

19.2

919

.53

19.7

720

.00

Tree Diameter Calibrations



Tree DiametersDirections: Measure the diameters of the trees at 6" above the ground.

Tree Number

Diameter of Tree Using Similar Triangle Calibrations

– Tree Stick

Diameter of Tree Using Diameter Tape

Average Diameter (of both methods)

1

2

3

4

5

6

7

8

9

10

| 27 |Forestry | Lesson 4

Forestry

Lesson Plan Tree Plot Study • Lesson 4

Learning ObjectivesStudents will be able to locate 1. trees in a coordinate system.Students will be able to create 2. a representative sample plot.Students will be able to apply 3. similar triangle and circle geometry to make tree measurements.

MaterialsTree measurement tools ❒(one per student)Compass (one per student) ❒

100' tape measure (one per team) ❒

Plot marking materials (at least 6 ❒per team) – these could be wood stakes, large nails, plastic cones or other highly visible and stable materials. Be careful about using pointed objects (e.g., wire survey flags) as marking materials as eye damage is a possibility.Lengths of 21' rope or ribbon of two ❒different colors(2 lengths per team)Worksheets (one of each ❒worksheet per student)

PreparationBe comfortable with plot studies •and coordinate systems.Be comfortable with compass use.•Locate a place and make •arrangements for students to visit a location with trees for students to measure. This should be a forested setting (not individual trees) that provides easy access and maneuverability in the forest. Students will be establishing 1/100th acre plots. These plots should contain between 4 and 10 trees of at least 4" diameter at 6" above the ground.Set boundaries for plot to be studied.•

Hour 1(20 min) Introducing sampling and plot study(40 min) Introducing coordinate systems

Hour 2(30 min) Developing a survey protocol(30 min) Laying out the plot coordinate system

Hour 3(60 min) Setting up the plot study

Hours 4 and 5(45 min) Review data to be collected and methods.

Conduct a sample study on a single tree.(75 min) Conducting the plot study.

1. IntroductionTell students that they now have the skills to determine how •many trees they need to fulfill their annual use of forest products. The next step is to conduct a study in a forest to determine how many trees it will take to total 75 cubic feet of forest products. One thing they will want to be able to do with the information is to determine how much area those trees cover (so not just knowing how many trees, but how much land is required to grow those trees). Ask students if they know the common unit of land •measurement (acre). Do any of them know how large an acre is? (43,560 square feet, about the size of a football field). The study they will conduct will help them determine how many trees there are on each acre of the forest where they do the study. Ask the students how they will determine that. Will they count all the trees on one acre? Why? (Too many trees, hard to keep track, take too much time). Instead they will take a sample of an acre. Discuss with students the benefits (time) and challenges (making sure the sample is representative, getting enough samples to minimize outlying data) of doing samples.Tell students that they will be using 1/100th acre •plots to locate and measure trees in the forest. Help the students calculate the dimensions of a square 1/100th acre plot. (Approximately 21' × 21') They will need calculators to use the square root function.


2. Coordinate system One of the things students will do in their plot samples is to locate the specific location of •individual trees. This will be done using a coordinate system. Introduce the coordinate system to students. An effective way to do this would be to tape X and Y axis lines to the classroom floor or in the playground or parking lot and place chairs at different locations. Students would then measure distances from each of the axes to plot the coordinates of the chair. Students can practice using the “Coordinate System” worksheet provided. Review the answers with students.Laying out the plot coordinate system. The plots will all be connected. The overall •design is flexible, but the layout should be a square or rectangle. As a conclusion to the project, a mural of the plot maps will be made by taping all of the individual plot maps together. An effective way to set up the plots will be to orient the X axes to north-south. This will allow a compass to be used to locate the coordinates of each tree. The X and Y axes would be in the middle of each plot. Each quarter would be 10'5" × 10'5". It is recommended that a map of the plot locations be drawn by the students. This will ensure •that each team locates the correct plot and that the relation to the other plots is understood. The number of plots needed will be determined by the number of teams formed.This is an example plot layout. Dashed lines represent plot X and Y axes.•

3. Developing a plot sample protocol. Share with students the information that will be collected in each of the plots: tree location in the •plot (coordinates), tree species, tree diameter, tree height. Tell the students that they should use the technique that they found the most accurate for finding tree diameter and height. Only trees larger than 4" in diameter will be located and measured. Ask students what process/steps should be used to collect that information. Work with the students to develop a protocol that will assure the information is collected. Part of the discussion should be how many students will be working in each plot (2-3 is recommended given the plot size). Also determine what tools will be used in collecting the data. Select/assign teams and assign each team a plot number. It is recommended to give higher achieving students the end plots as these have the least overlap with other plots and therefore there is less guidance in establishing those plots and they also help to orient the rest of the plots.

4. Setting up the plots. Take students to the identified forest to set up the plots. If necessary, provide students •instruction on compass use so they can at least accurately locate north, south, east and west. (For background on compass use, see the www.learn-orienteering.org, lesson on compass use www.learn-orienteering.org/old/lesson1.html)

http://www.learn-orienteering.org

http://www.learn-orienteering.org

http://www.learn-orienteering.org/old/lesson1.html


Have students locate the corners of their 1/100th acre plots. The sides should run directly north-•south and east-west respectively. Each plot will share at least one side (many will share 3 sides) with a neighboring plot. Neighboring plot teams should consult each other in the alignment of the edges of the plots – the plot edges and corners should meet and therefore share corner stakes. Once the plot corners are located, the students should locate and mark their plot X and Y axes using rope or ribbon. To do this, have students locate the mid-point of each side of their plot. An easy way to find the midpoint would be to find where the diagonal lines from opposite corners intersect. Again, these mid-points should match the neighboring plots so students should work cooperatively to locate these points. Run a rope or ribbon between these points. These are the X and Y axes of the plot. All trees will be located in reference to these axes in a coordinate system.If necessary, this is a good time for students to learn tree identification skills. •Complete materials for doing a tree identification lesson are available on the LEAF Program web site at www.uwsp.edu/cnr/leaf/treeid.shtml

5. Conducting the plot study. Review with the students the information they will be gathering in each of their plots (tree •species, tree diameter, tree height, and tree location) and the protocol they developed earlier. One note about tree height: In the earlier activity, students measured the total height of •the tree. Since we are concerned with the amount of forest products, students should be measuring the usable/merchantable height of the tree. This requires measuring the tree height at a point of about 4" in diameter. Obviously this is an estimate because it can’t be directly measured. Help student determine where 4" would be near the top of the trees.To locate the coordinates of each tree, students will have to work in pairs. They will use a compass to •sight north-south and east-west to determine where the tree aligns on the X and Y axes, respectively.For example, for the tree in the plot below, students would sight directly north to see where •the tree aligns on the X axis and directly east to see where the tree aligns on the Y axis. One student would stand at the tree and direct a student on the X axis to the point where the tree

aligns on the axis. This would be repeated for the Y axis.Go through the data to be collected using a sample tree outside of the plots. Remind students they will only be measuring trees that are more than 4" in diameter at the base.

Have students work to identify, measure, and locate each of the trees in their plot. Have student use the “Plot Study Data sheet” to record their data.

Pictures of each tree could be taken for data collection and shown during the wrap up activity.

6. Wrap upWhat did your students find? Was there one species of trees that dominated their plots? •Was diameter or height of the trees uniform? Were there more trees in one spot or were they evenly distributed? Have them share their maps that they created of their plot.

Example Plot 20' 10" × 10' 10'

http://www.uwsp.edu/cnr/leaf/treeid.shtml



Coordinate System

b

a

d

1. Provide the coordinates of the points Point X coordinate Y coordinate

a

b

c

d

2. Place the following points by coordinates Point X coordinate Y coordinate

e -3 -1.5

f 1 -2

g 0 2

h -2 2

c

Y

X



Plot Study Data sheet

Tree Number Tree Species Tree Diameter

(inches)

Tree Diameter(convert to

feet)

Tree Height (to 4" top)

(feet)

X coordinate(feet and inches)

Y coordinate(feet and inches)

1

2

3

4

5

6

7

8

9

10


Forestry

Lesson Plan Using the Data • Lesson 5

Learning ObjectivesStudents will be able to use a 1. formula to calculate tree volumes using diameter and height data.Students will be able to translate 2. data to a per acre basis.Students will be able to 3. able to graph tree data.Students will be able to calculate 4. basic probability data.Students will be able to use scale 5. and mapping techniques to create a drawing of their plot.

MaterialsWorksheets (one of each ❒worksheet per student)Example maps (multiple) ❒

Large sheets of paper (one per team) ❒

Colored pencils or other ❒coloring, drawing materials

PreparationBe comfortable with the ❒volume calculations, graphing, basic statistics, basic probability, and mapping.

Hour 1(60 min) Calculating tree volumes

Hour 2(45 min) Graphing plot data15 minutes Translating data to a per acre basis

Hour 330 minutes Introducing basic statistics and probability 30 minutes Calculating statistics and probability of tree plot data

Hours 4 and 530 minutes Introducing mapping techniques and scale90 minutes Mapping plots

Lesson Extensions

Students could investigate finding the volume of different •cones and/or frustum of a cone. Use ice cream cones filled (by measuring out) with ice cream or whipped cream. Have the students find the volume of a traffic cone filled with water. Put a little in at first and see how the volume fills up quite quickly in the narrow parts and slowly in the wider parts. This would help illustrate how finding the volume of a shape that doesn’t have a constant diameter involve more complicated math.The data that the students collect will be all numerical. •You could have your students take their calculated volumes, heights, or diameters and do one of the following: frequency tables, histograms, circle graphs, stem and leaf plots, box-and-whisker plots, scatter plots, and mean, median, mode and range. Instructions for how to do these graphs, go to http://www.purplemath.com

http://www.purplemath.com


1. Tree volumesTrees are shaped like cones that don’t come to a point. This shape is called a frustum of •a cone. The equation to determine the volume of a frustum is V=

€

πh3

(R2 + r2 + r)Foresters use this equation to calculate volume because their •

tape only measures diameter: V =

€

πh12 (D2+D•d+d2) where

h=height, D=bottom diameter, d=top diameter, π=3.14. One of the tricky parts of the calculation is that the •

diameters are measured in inches, where the height (and desired volume) is in feet. Therefore, the diameters must be converted to feet before doing the calculation. The top diameter (d) will be 4" = 0.333 feet. The bottom diameter (D) will be the

diameter measured by the students. The height (h) is also determined by the students.Work with the students to understand the formula. For most students, this will •be a difficult formula. There are multiple options to complete the calculations. Students can do it by hand, set up a spreadsheet with the formula, use an on-line volume calculator (e.g., www.analyzemath.com/Geometry�calculators/surface�volume�frustum.html), or use the “Tree Volumes Table” to find the volumes.However the students calculate the volumes, have students determine the volume of each •tree in their plot. Assign the pairs of students from each plot a number. Record the volumes in the “Tree Volumes” data sheet and have students sum the volumes in their plot.Have students create graphs using the data they collected in their plots. A wide variety of •graphs can be constructed that will help students understand the data. Potential bar graphs would include number of trees by species, number of trees by diameter, number of trees by height, height of tallest tree by species, and diameter of largest tree by species. Pie charts could be made of the number of trees by species. Make sure they label their graphs.If desired, spreadsheet software could be used to create the graphs. Examples are below. •Have students share their graphs. Discuss with students what they can •interpret from the graphs. Are there differences between the plots?

Now that students have the volumes of the trees in their plots, this can be related to a •per acre basis. Ask students to recall how large of an area they sampled (1/100th of an acre). Then ask them what they need to do (multiply by one hundred). Have students do these calculations and record them on the ‘Tree Volume per Acre” data sheet. Have each team share their data with the others in the class and then calculate the average across all teams. Discuss with students. What is the average volume per acre?Introduce basic statistics and probability. Students will use mean, median, mode, •

D

d

h

http://www.analyzemath.com/Geometry_calculators/surface_volume_frustum.html

http://www.analyzemath.com/Geometry_calculators/surface_volume_frustum.html


range, and percentages to understand the data that they have collected. Use the “Statistics and Probability Worksheet” to introduce these statistics.Have students calculate the mean, median, mode, and range of their plot data tree heights, tree •diameters, and tree volumes. Have students calculate the probability of finding a specific kind of tree for each of the tree species. Have students put all of their plot data and statistics on the board or some other accessible/visible place (such as Excel and then projected onto the wall) so that a full class data set can be created. If desired, the same statistics and probability calculations can be done with the entire data set so that it can be compared with the individual plot data sets.Students can use the “Plot Sample Summary” sheet to summarize the data. •There are questions about the diameter of the trees. Generally, low average diameter (e.g., less than 10”) indicates a young forest. A wide range of diameters represents an uneven-aged forest where a narrow range represents an even-aged forest.Have students compare and contrast their plots with the other plots. Is there •variability across plots? What does this mean about sampling? (Generally, the more samples the better to get a better picture of the whole.)

2. MappingIntroduce mapping by showing the students a variety of maps. Ask them what maps are •(representation of some real world features). Discuss with students the common features of the maps. Important components of each map include: title, aspect (directional orientation), legend/key with symbols, and scale. Explain to the students that they will be making a map of their individual team plots that indicate the location of each tree which will be put together with the other plot maps to make a larger map of all of the plots. Therefore, it will be essential for the students to be accurate and all use the same map features. Brainstorm with the students the goal of the map, appropriate scale, symbols, aspect, and other map features. The goal should be to indicate the location of each tree in the plot and provide information about species and size of each tree. The students could determine whether the size is best represented by volume, diameter, and/or height. The scale will be determined by the size of the paper. The map should take up as much of the paper as feasible and include landforms.Provide the teams with the paper and equipment to create their maps. •



Mer

chan

tabl

e H

eigh

t (fe

et)

Dia

met

er

(inch

es)

1015

2025

3035

4045

5055

6065

7075

8085

9095

100

40.

91.

31.

72.

22.

63.

13.

53.

94.

44.

85.

25.

76.

16.

57.

07.

47.

98.

38.

7

51.

11.

72.

22.

83.

33.

94.

45.

05.

56.

16.

77.

27.

88.

38.

99.

410

.010

.511

.1

61.

42.

12.

83.

54.

14.

85.

56.

26.

97.

68.

39.

09.

710

.411

.111

.712

.413

.113

.8

71.

72.

53.

44.

25.

15.

96.

87.

68.

49.

310

.111

.011

.812

.713

.514

.415

.216

.016

.9

82.

03.

14.

15.

16.

17.

18.

19.

210

.211

.212

.213

.214

.315

.316

.317

.318

.319

.320

.4

92.

43.

64.

86.

07.

38.

59.

710

.912

.113

.314

.515

.716

.918

.119

.320

.621

.823

.024

.2

102.

84.

35.

77.

18.

59.

911

.312

.814

.215

.617

.018

.419

.821

.322

.724

.125

.526

.928

.3

113.

34.

96.

68.

29.

911

.513

.214

.816

.518

.119

.721

.423

.024

.726

.328

.029

.631

.332

.9

123.

85.

77.

69.

511

.313

.215

.117

.018

.920

.822

.724

.626

.528

.430

.332

.134

.035

.937

.8

134.

36.

58.

610

.812

.915

.117

.219

.421

.523

.725

.928

.030

.232

.334

.536

.638

.840

.943

.1

144.

97.

39.

712

.214

.617

.119

.521

.924

.426

.829

.231

.734

.136

.539

.041

.443

.846

.348

.7

155.

58.

210

.913

.716

.419

.221

.924

.627

.430

.132

.835

.638

.341

.043

.846

.549

.252

.054

.7

166.

19.

212

.215

.318

.321

.424

.427

.530

.533

.636

.639

.742

.845

.848

.951

.955

.058

.061

.1

176.

810

.213

.617

.020

.323

.727

.130

.533

.937

.340

.744

.147

.550

.954

.257

.661

.064

.467

.8

187.

511

.215

.018

.722

.526

.230

.033

.737

.541

.244

.948

.752

.456

.259

.963

.767

.471

.274

.9

198.

212

.416

.520

.624

.728

.832

.937

.141

.245

.349

.453

.557

.661

.865

.970

.074

.178

.282

.4

209.

013

.518

.022

.527

.131

.636

.140

.645

.149

.654

.158

.663

.167

.672

.176

.681

.285

.790

.2

219.

814

.819

.724

.629

.534

.439

.344

.349

.254

.159

.063

.968

.873

.878

.783

.688

.593

.498

.4

2210

.716

.021

.426

.732

.137

.442

.848

.153

.458

.864

.169

.574

.880

.285

.590

.996

.210

1.6

106.

9

2311

.617

.423

.229

.034

.740

.546

.352

.157

.963

.769

.575

.381

.186

.992

.698

.410

4.2

110.

011

5.8

2412

.518

.825

.031

.337

.543

.850

.056

.362

.568

.875

.081

.387

.693

.810

0.1

106.

311

2.6

118.

812

5.1

2513

.520

.226

.933

.740

.447

.153

.960

.667

.474

.180

.887

.694

.310

1.0

107.

811

4.5

121.

212

8.0

134.

7

2614

.521

.728

.936

.243

.450

.657

.965

.172

.479

.686

.894

.110

1.3

108.

511

5.8

123.

013

0.2

137.

514

4.7

2715

.523

.331

.038

.846

.554

.362

.069

.877

.585

.393

.010

0.8

108.

611

6.3

124.

113

1.8

139.

614

7.3

155.

1

2816

.624

.933

.241

.449

.758

.066

.374

.682

.991

.299

.510

7.8

116.

112

4.3

132.

614

0.9

149.

215

7.5

165.

8

2917

.726

.535

.444

.253

.161

.970

.879

.688

.497

.310

6.1

115.

012

3.8

132.

714

1.5

150.

415

9.2

168.

017

6.9

3018

.828

.337

.747

.156

.565

.975

.384

.894

.210

3.6

113.

012

2.4

131.

814

1.3

150.

716

0.1

169.

517

8.9

188.

3

Tree Volume Table(cubicfeet)(assumingatopdiameterof4")



Tree VolumesRemember:Showyourworkandmakesureyouusethecorrectlabelonallyouranswers!

Tree Number Tree Species

Tree Diameter(inches)

Tree Height (to 4" top)

(feet)Volume (cubic feet)

1

2

3

4

5

6

7

8

9

10

Total Volume



Tree Volume per Acre

Tree Volume per Acre(cubic feet)

Team 1’s Data

Team 2’s Data

Team 3’s Data

Team 4’s Data

Team 5’s Data

Team 6’s Data

Team 7’s Data

Team 8’s Data

Team 9’s Data

Team 10’s Data

Average



Statistics and ProbabilityRemember:Showyourworkandmakesureyouusethecorrectlabelonallyouranswers!Roundtothenearest100th.

Mean = average of all the numbers• Median = the number that is in the middle of the range of numbers• Mode = the most frequently occurring number• Range = the distribution of the numbers• Experimental Probability = chance of something happening based on the data from an experiment•

Statistics

1. Using the data set of tree diameters above, calculate the following:Mean = ��a. Median = ��b. Mode = ��c. Range = ��d.

Probability

The experimental probability of any tree having a particular diameter is the percentage or ratio of a particular diameter to the total number of trees. For example, the experimental probability of a tree having a diameter of 19" is 1 out of 16, which is equal to 6.25% (1/16 × 100).

Find the probability of:

2. A tree having a 4" diameter: ��

3. A tree having a diameter greater than 20": ��

4. A tree having a diameter less than 10": ��

Tree Diameters (inches)12 19 6 12 4 13 4 15 23 27 7 5 9 4 7 8



Plot Sample SummarySummarize the data from all the plots in the table and in response to the questions below.

Statistic Diameter Height Volume

Mean

Mode

Median

Range

1. What was the most common tree species? ��

2. What is the probability of a tree being the most common species? ��

3. What is the probability of a tree being the least common species? ��

4. Based on the mean of all the plot data, how much land is required to grow 75 cubic feet of wood to meet your annual consumption?

5. What does the mean diameter tell you about this forest?



6. What does the range of diameters tell you about this forest?

7. Which statistic (mean, median, mode, or range) would be the best to describe each group of measurements (diameter, height, and volume)?

8. (Homework) Write a reflection on the differences among the data and what that could mean.


Forestry

Lesson Plan How Many Trees? • Lesson 6

Learning ObjectivesStudents will be able to use the 1. data that has been collected from a response to question posed at the beginning of this unit: How many trees do each of us use?Students will utilize effective 2. presentation techniques.

MaterialsPresentation materials ❒

Tape for maps ❒

Presentation rubric (one per team) ❒

PreparationReview the example “Final ❒Presentation Rubric” and make changes as desired. This will be used to guide the students’ presentations.Prepare copies of other team’s data. ❒

Hour 1(30 min) Review of information collected and setting

expectations for the presentations(30 min) Preparation for presentations

Hour 2 and 3(100 min) Student presentations(20 min) Conclusion

1. ReviewReview the information that has been collected and •calculated over the course of the unit: tree heights, diameters, tree species, tree volumes, plot tree volumes, tree volumes per acre, and a variety of statistics and probability. Work with students to brainstorm or set the topics to be covered in a final presentation of the information remembering that the main question of the unit was how many trees do each of us in the United States use annually. To answer this question, students should determine whether they will use information from their team’s plot or the combined teams’ data and justify why they made that decision (it really should be done with the entire group data as this will be the most accurate given multiple samples). The length of the presentations is also optional, but approximately 10 minutes should be sufficient. Students should provide highlights of data collected and calculations done and should explain their map to rest of the class. A poster or PowerPoint presentation may be appropriate to present the data. An “Example Final Presentation Rubric” is provided. Teachers may add points to the rubric as they see fit. Bonus points may be awarded if they address the central question of the unit.

2. ConclusionFollowing the presentations, review again with students •what they learned. Ask students for their perspective on how much wood we consume. Is it too much? Are there any drawbacks to using that much wood? Are there benefits?


Name ��Final Project

Final Presentation Rubric

Team/Students: ��

Quality

Presentationfeature Exceptional Adequate Minimally

accomplished Notaddressed

Howmanytreesdoweuse

Provides the number of trees required to meet annual use of forest products and the area needed to grow those trees.

Provides statistics to support calculation. Provides personal

perspective on their reaction to

consumption.

Provides the number of trees required to meet

annual use of forest products and

the area needed to grow those trees. Provides some personal perspective on

their reaction to consumption.

Provides limited data based on team’s plot. Only includes

number of trees.

Not mentioned

Useofstatistics

Includes required statistics and

provided analysis of the meaning of the statistics.

Includes statistics required.

Includes some of the required statistics. Not included

Map

Map is highly detailed, creative,

and neat. Explanation is

complete, accurate, and provides insight

into meaning of the features.

Map includes all of the required

features. Explanation of the map is complete

and accurate.

Map includes most of the required

features. Explanation of map is incomplete

or inaccurate.

Map not completed.

Presentation

Presentation is given with interest, good eye contact,

appropriate volume and includes creative methods and tools to show data and map.

Presentation is given with interest, good eye contact,

appropriate volume, and

effective methods of showing data

and map.

Presentation is completed but lacks critical components.

Not given.

Forestry | Unit 6 | 43 |

Unit Answer KeysSomeoftheanswerswillbeshowninstepstoguideinstruction.Thestepswillbeseparatedbyarrows.

Lesson 1Skill Building: How Much is 75 Cubic Feet?

Liters of soda: 2,142 bottles1. Sheets of paper: .0002 ft2. 3; 333,777.18 sheets of paperCell Phones: Answers will vary by phone3. Textbook: Answers will vary by textbook4. Locker: Answers will vary by locker5. Gallon of Milk: 560.96 gallons of milk6.

Lesson 2Skill Building: Triangle Angles and Sides

601. º

602. º

603. º

1804. º

2 5. 1∕8"2 6. 1∕8"2 7. 1∕8"668. º

489. º

6610. º

18011. º

2"12. 2"13. 2 14. 5∕8"

4515. º

9016. º

4517. º

18018. º

2"19. 2"20. 2 21. 7∕8"

Skill Building: Similar Triangles

401. º

902. º

503. º

2 4. 9∕16"2 5. 1∕8"3 6. 5∕16"407. º

908. º

509. º

1 10. 1∕4"1 11. 1∕16"1 12. 5∕8"2 13. 9∕16": 1 1∕4"2 14. 1∕8": 1 1∕16"3 15. 5∕16": 1 5∕8"

2216. º

11017. º

4818. º

4 19. 9∕16"2 20. 5∕16"5 21. 3∕4"2222. º

11023. º

4824. º

1 25. 1∕2"3∕4"26. 1 27. 15∕16"4 28. 9∕16": 1 1∕2"2 29. 5∕16": 3

∕4"5 30. 3∕4": 1 15∕16"

Skill Building: Right Triangles

301. º

602. º

903. º

2 4. 1∕2"1 5. 1∕4"2 6. 3∕16"107. º

908. º

809. º

2 10. 7∕8"911. ∕16"2 12. 15/16"9013. º

3714. º

5315. º

2 16. 7∕16"3 17. 1∕16"1 18. 7∕8"9019. º

4520. º

4521. º

2 22. 5∕16"3 23. 1∕4"2 24. 5∕16"

Skill Building: Using Similar Triangles

Triangle 2Side AB: 7.69"1. Side BC: 6.41"2.

Triangle 43. Side EC: 1.52"4. Side DC: 1.22"

Skill Building: Using Isosceles Right Triangles

Angle A = 901. º

Angle B = 452. º

Angle C = 453. º

BC = 108.9'4. Tree= 41'5.

Skill Building: Similar Triangle Search

1, 101. 2, 32. 4, 53. 6, 94. 7, 115. 8, 126.

Forestry | Unit 6| 44 |

Lesson 3Skill Building: Calculations for Construction

Circle 1Diameter: 5.4 cm1. Radius: 2.7 cm2. Circumference: 16.9 cm3. Area: 22.9 cm4. 2

Circle 2Diameter: 2.5 cm5. Radius: 1.3 cm6. Circumference: 8 cm7. Area 5.3 cm8. 2

Circle 3Diameter: 25.4 cm9. Circumference: 79.8 cm10. Area: 2,025.8 cm11. 2

Circle 4Diameter: 17.8 cm12. Radius: 8.9 cm13. Area: 248.7 cm14. 2

Lesson 4Skill Building: Coordinate System

1. Provide the coordinates of the points Point X coordinate Y coordinate

a -1 -1

b -3 0

c 2 3

d 0 -3

2. Check to make sure that points are plotted correctly on the coordinate system

Skill Building: Plot Study Data sheet

Answers will vary by student

Lesson 5 Skill Building: Statistics and Probability

Mean: 10.91. Median: 8.50

Mode: 4 Range: 23

18.75%2. 12.50%3. 56.25%4.

Forestry | Unit 6 | 45 |

Unit ReferencesLearn Orienteering, How to Use a Compass Guide – lessons on compass use 1. www.learn-orienteering.org/old/

LEAF Program – lesson guides and resources about Wisconsin forests 2. www.uwsp.edu/cnr/leaf

Private Forest Management, Auburn University – “Measurement of 3. Tree Heights” www.pfmt.org/inventories/height.htm

UW-Extension, Forestry Facts Publications – background information on forestry mea-4. surements and forest management http://forest.wisc.edu/extension/forfact.htm

Strategies for Dealing with Similar Triangles 5. http://regentsprep.org/Regents/Math/similar/Lstrategy.htm

http://www.learn-orienteering.org/old/

http://www.uwsp.edu/cnr/leaf

http://www.pfmt.org/inventories/height.htm

http://forest.wisc.edu/extension/forfact.htm

http://regentsprep.org/Regents/Math/similar/Lstrategy.htm

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