Students should already be familiar with the following concepts and vocabulary:
- diameter and radius
- pi (π)
Prior to this lesson, the teacher (with or without students) should
identify the trees in their schoolyard or surrounding community that
will be measured, and check that the species is listed on one of the
tree tables provided. Borrow a tree guide from your library or use the
online guides, such as Tree Identification Guide at arborday or What Tree Is It?.
The tables provided in this lesson are based on data taken from
trees in Illinois. If you are in a different climate, be sure to
explain to the students that the estimation will be less accurate. The
second Lumberjack Table
is based on urban grown trees, while the Ranger Table and the first
Lumberjack Table are based on forest grown trees. (Urban trees grow
faster because they usually have better care and have more sunlight
than those in the forest.) You may want to try to contact your local
arboretum for more accurate information on trees in your community.
Use tree markers to record the name of the species for each tree
identified. If there is two of the same type, label them with a number
(e.g. oak 1, oak 2). Unbend wire hangers to push into ground near tree
Prepare materials in advance. Each group will need one laminated
tree table, yardstick, 4 yards of string, activity sheet, marker,
calculator, and pencil. Decide if you want the students to work outside
after measuring, or return to the classroom to do the calculations. If
working outdoors, each group should take a clipboard. Print overheads
on transparencies if needed.
Choose the activity sheet and table you will use with your class based on the grade level or abilities of students. The Ranger Talk activity sheet and Ranger Table
are designed for grades 3-5. In this simpler task, once students have
determined the diameter, they need to read the growth rate factor for
their species on the table and multiply it by the diameter to calculate
Lumberjack Talk activity sheet and Lumberjack Tables
are for grades 6-8. These tables list diameters by multiples along with
corresponding ages of the trees. If a measurement falls in between the
groupings listed, students will have to do additional calculations to
determine the age.
Explaining the Activity
Introduce the lesson by allowing students to talk about trees that they
have in their yard or on their street. Ask if they ever wondered how
old some of the trees are. Explain that scientists use different ways
to calculate the age. Allow students to volunteer any prior knowledge
of how this is done. If not, talk about growth rings and core samples.
Bring in a cross section of a tree if one is available or share the Tree Rings overhead.
There are three ways to determine the age of a tree. The most
accurate way is to count rings, but only if the tree has been cut down.
Core samples can be taken from live trees, but the process can cause
damage to the tree. The third way uses math. Scientists have created
tables based on the growth rate of trees. Given the diameter of a tree,
the tables can be used to estimate the age of a tree.
Ask students why it would be difficult to measure the diameter of
the tree. [You can’t put a ruler or tape measure through the tree.] For
lower grade students discuss the definition of π. It is the ratio of a
circle’s circumference to its diameter. The value of π is a little more
than 3. Review the formula for circumference, as well. C= π × diameter.
Lumberjacks would only need reminding to change the circumference
formula to solve for diameter, giving D = C ÷ π.
For rangers, use an example to help students understand why D = C ÷
π. Ask, “What happens if you know the circumference, but not the
diameter?” Round π to 3 to simplify the problem so students can focus
on the algebra. Show students the equation 15= 3 × □. What is the
relationship between the 15 and the 3 that could give you an answer?
[15 ÷ 3 = 5] This should help students see that to find the diameter
they can always divide the circumference by π. So diameter =
circumference ÷ π.
For both levels, model the use of the formula with a circumference
of 40. Plug the quantity 40 in to the formula D=C ÷ π. It would be good
practice for students to estimate first, using the approximation π ≈ 3.
With a circumference of 40, D ≈ 40 ÷ 3 ≈ 39 ÷ 3 = 13. This will show if
the answer is reasonable. Next use a calculator to find the actual
answer and round to the nearest whole number. When using calculators,
rangers should use 3.14 as pi because it may be easier for them to work
with fewer decimal places when rounding. As you work through the
example, you may want to review rounding to the nearest whole number,
as they will need to do this to record their diameters. Lumberjacks
should use the π button on their calculators.
Next, take the chosen table and display it on an interactive
whiteboard or overhead projector. Use the diameter of a pin oak tree as
The Ranger Table shows the pin oak has a growth rate of 3. By
multiplying the diameter times the growth rate, you can get the
estimated age of the tree from the example above. 13 × 3 =39 years old
The Lumberjack Urban Table only lists diameters in multiples of 5.
13 inches is between 10 and 15. The age for pin oak is between 27 and
38 for a diameter of 13 inches. For an accurate calculation, set up a
proportion. 10 is to 13 as 27 is to what? (10/13 = 27/n cross multiply
and solve for n) [35 years old] For a quick estimate show students the
following: the middle of 27 and 38 is 32.5. Since 13 is a little more
than half of 10 -15, the answer would be a little more than 32.5, maybe
around 34 years old.
Divide the class into groups of 3 or 4 students. Small groups will
allow all students to be involved. Distribute activity sheets and have
students gather materials needed for their group. For the Lumberjack
Activity Sheet, tell the class which table they will be using and have
them record it on their paper. Take the class out to the school yard to
one of the trees listed on the table. Model the activity with the whole
group, before sending them off to collect their own data. Using a piece
of string, have a few children work together to measure the
circumference of the trunk. The proper height to measure at is 4 feet.
For younger students, that would be at about their eye level. Middle
school students can measure at their shoulder height. Tell them that it
is important that the string be at the same height around the tree to
get the most accurate reading. (You may want to bring a 4 ft. piece of
string for each group to use as a check for the height of the
measurement of the circumference. Mark the measurement on the string
with a marker. Choose two other students to use a yard stick to measure
the string to the mark. You may have to review with the younger
students what to do if the string is longer than the yardstick. Record
the measurement on a small whiteboard. Choose another student to use
the formula and find the diameter (using the calculator). Round to the
nearest whole number and determine the age of the tree as modeled in
the classroom using the Tree Table.
Assign two of the previously identified trees to each group to
measure. Calculations can be done outdoors or back in the classroom.
Groups record their data on the Ranger Talk or Lumberjack Talk
overhead for all to see. Which tree was the oldest? Which tree was the
youngest? Looking at the master sheet, was the oldest tree the one with
the longest diameter? If not, why? [Different trees grow at different
rates.] Using the Ranger Table, ask the students if the growth rate is
larger, does that mean that type of tree grows faster? [The higher the
growth rate, the slower a tree grows!] You may want to clarify this by
comparing a dogwood to an aspen. If a dogwood diameter was 1 inch, it
would be 7 years old: 1×7. An aspen that measured 1 inch would only be
2 years old: 1×2. A dogwood only grows 1/7 of an inch in one year. An
aspen grows ½ of an inch in one year.
Using either of the Lumberjack Tables, ask students if the trees
seem to grow at the same rate no matter what their age? How could you
prove that? [A particular tree does not grow at the same rate
throughout its life. If you graph the years versus diameter, you will
see that it is not a straight line.]
Questions for Students
1. Why can we look at the size of a tree’s diameter and know the age?
[Each year a tree gets wider because it adds a growth ring. The average width of a ring is determined by the type of tree.]
2. If different groups measured the same tree and got different results, how could that be possible?
[Answers will vary. Some possible answers include: Students were
measuring at eye level, and they may be different heights. Students did
not have the string pulled tightly around the trunk.]
3. If you were given the age and growth rate of a tree that is not on
the list, how would you be able to determine its diameter? How could
you find its circumference?
[Divide the age by the growth rate to find
the diameter. Multiply the diameter times π to find the circumference.]
4. Using diameter to find the age of a tree is only an estimate.
Discuss what types of things might affect growth of a tree from year to
[Amount of rain, fire or other damage, amount of sunlight.]
5. Explain this statement: “Measurements are approximations and differences in units affect precision.” Do you agree or not?
[It is very difficult to be exact when measuring. You may have
difficulty looking right on and if you looking at a ruler a bit to the
left, it will change the results. If you round to the nearest cm.
versus nearest inch, that will change your measurement. ]
- Were the students engaged in this lesson? Could you have done anything differently to make them more engaged?
- Was classroom management an issue? Can you think of how the lesson
could have been structured differently to make it work for your class?
- Did students work productively in their groups?
- Do you find it beneficial to use real life math lessons in your classroom?
- Measure circumference
- Compute diameter when circumference is known
- Use a table to determine best estimation of tree age
Common Core State Standards – Mathematics
Grade 3, Measurement & Data
Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l). Add, subtract, multiply, or divide to solve one-step word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.
Grade 3, Measurement & Data
Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units-- whole numbers, halves, or quarters.
Grade 4, Measurement & Data
Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale.
Grade 5, Geometry
Classify two-dimensional figures in a hierarchy based on properties.
Grade 7, Geometry
Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.
Grade 7, Stats & Probability
Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.
Grade 7, Stats & Probability
Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.
Grade 8, Stats & Probability
Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
Common Core State Standards – Practice
Make sense of problems and persevere in solving them.
Model with mathematics.
Use appropriate tools strategically.
Look for and make use of structure.