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Students will prepare for measuring things which they cannot reach, using concepts of measurement, geometry, and proportion.


The teacher will read aloud The Librarian Who Measured the Earth, by Kathryn Lasky, and students will have access to other reading materials such as Sir Cumference and the First Round Table, and other books related to measurement, geometry, and proportion.


Circumference/Diameter/pi Activity

Students participate in a discovery activity which demonstrates the relationship between diameter and circumference.  Students are provided with a collection of cylinders of different sizes (e.g., cardboard tubes of different sizes from toilet paper, paper towels, wrapping paper; lids or plastic containers, etc.). 

1.     Pairs of students are asked to measure the diameter of one cylinder, marking its length on a string with a pen.

2.     Next, the student team measures around the cylinder with the diameter-marked string, making a mark at each unit of diameter length.

2.     Students write down how many diameter lengths make up the circumference.

3.     Students do this for each cylinder in their kit.

4.     Students will discover that regardless of the size of the cylinder, its circumference will be a little bit more than three times its diameter.


Students can actually measure the diameters and circumferences using a cm or inch tape measure, and calculate a more exact relationship for each cylinder, recording them on a data sheet.  Results should demonstrate a close proximity to pi for each cylinder.


In subsequent group discussions, students devise a formula to represent the relationship they discovered, with Õ representing the repeating factor they discovered in their cylinder activity.  The formula should look something like:



Students can confirm their findings by referring to math text explanations of pi, and practice circumference/diameter problems from the text using pi.


Height Measurement

Students practice both methods of height measurement (see description of methods under Field Site Activites, below) on the school site, beginning with structures of known height and working toward objects of unknown height.  Students compare their results using both methods.


In class discussions, students discuss both methods, the benefits of having two or more methods to confirm a particular measurement (no answer key in life!!), and conjecture about the circumstances that would lead one to select one method over another if both were not possible.  



            The Librarian Who Measured the Earth°, Kathryn Lasky

            Sir Cumference and the First Round Table,

            Math textbook

            Yard or meter sticks

            Tape measure



Students will be evaluated on their participation in the pi discovery activity and subsequent discussions.  Data sheets and circumference math problems confirm the studentšs ability to measure, to discern patterns, and to understand the relationship between pi, diameter, and circumference.



Students pairs measure the size of a tree by measuring its trunk, crown (broadest width of tree including branches), and height.  Students record their data on a data sheet at the site.  Before the field trip, students will have:

ˇ       learned vocabulary words horizontal, vertical, measure, crown, trunk, circumference, diameter, round (math), straight, opposite, length, base, and height

ˇ       read and understood activity instructions

ˇ       studied the relationship between circumference and diameter (d×p = C)

ˇ       practiced the height measurement method.


Student Instructions:

Instructions for measuring the trunk:

1.     Measure from the ground to 4 1/2 feet high on the trunk.

2.     At that height, measure the trunk's circumference.  Use a string around the trunk and measure the length of the string.

3.     Round to the nearest inch. Record the number as the circumference.  Make sure you and your partner agree on the results.

4.     Using a calculator, calculate the treešs diameter (circumference ¸ p), round to the nearest tenth of an inch, and record.  Make sure you and your partner agree on the results.


Instructions for measuring the crown:

1.     Find the tree's five longest branches.

2.     Put markers on the ground beneath the tip of the longest branch.

3.     Find a branch that is opposite it and mark its tip on the ground.

4.     Measure along the ground from first marker to the second marker.

5.     Record the number and label as crown.  Make sure both you and your partner agree on the results.


Instructions for measuring the height - Method I:

1.   Have your partner stand at the base of the tree.

2.     Back away from the tree, holding your ruler in front of you in a vertical position.  Keep your arm straight.  Stop when the tree and the ruler appear to be the same size. (Close one eye to help you line it up.)

3.     Turn your wrist so that the ruler looks level to the ground and is in a horizontal position.  Keep your arm straight.

4.     Have your partner walk to the spot that you see as the top of the ruler.  Be sure the base of the ruler is kept at the base of the tree.

5.     Measure how many feet he or she walked.  That is the tree's height.  Round to the nearest foot and record your answer as the height.

6.     Switch roles with your partner and repeat Steps 1-5.  If you and your partner obtain completely different results, measure the height again until you agree on an accurate measurement.


Instructions for measuring height ­ Method 2 (best done close to midday when shadows are shortest):

1.     Measure the length of the shadow of a tree.

2.     Measure the length of the shadow of an object of known height (e.g., the student)

3.     Have students establish an equation representing equivalent proportions between shadow length and height.

a.     tree height ÷ tree shadow length = student height ÷ student shadow length

4.     Have students solve for tree height (use calculator)

a.     tree height = tree shadow length x student height ÷ student shadow length



Materials and resources:

String, 12-inch ruler, paper, pencil, data sheet, yard stick or tape measure, calculators, tree



If possible, allow time for pairs in each group to compare answers and then remeasure the tree if needed.  When independently derived answers are in agreement, students will make the assumption their measurements are correct.  If numbers do not agree, and time does not allow, students should be able to hypothesize some reasons for differences.  Student product for formal evaluation:  data sheets.





Students discuss their experiences of measuring the treešs circumference, diameter, height, and crown.  In a class discussion, they compare and analyze their results, and suggest possible reasons for variations or error.  Students relate their experiences to the more general topic of measurement, discussing issues relating to accuracy (ballpark v. pinpoint accuracy), methodology, practicality, etc., and relate these issues to other kinds of measurement in the world.


A guest speaker (e.g., forestry resource manager) comes to the classroom to talk about how trees are measured (for purposes of resource management, dendrochronology, fire prevention) in real-life applications.  The speaker relates the kinds of measurements the students made with these real applications.  The speaker also introduces students to possible career paths in forestry and resource management.


                        MATERIALS AND RESOURCES

Speaker (contact local environmental education resources, USDA Forestry Dept., state and county forestry resources, private timber companies who employ resource managers, etc.)



Informal evaluation is based on student participation in discussions and speaking events.




° California Department of Education Recommended Literature