## Tree Talk

- Lesson

If a tree could talk, we could ask it how old it is. Here is a mathematical way to estimate the age of your schoolyard trees. Students will measure circumference of trees in order to find diameter and calculate age of local trees using a growth rate table.

Students should already be familiar with the following concepts and vocabulary:

- diameter and radius
- circumference
- 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 Arbor Day" or "What Tree Is It?". Perform a simple internet search (with quotation marks) to find these sites.

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 after labeling.

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 the age.

Lumberjack Talk Activity Sheet and Lumberjack Tables are for grades 6–8.

Lumberjack Talk Activity Sheet

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 an example.

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.

**Activity**

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.]

### References

- Dwyer, J. What are the Ages of Trees in Your Neighborhood? Retrieved from http://www.mortonarb.org/tree-plant-advice/article/16371/what-are-the-ages-of-trees-in-your-neighborhood.html.
- Jones, T., Bowles, M., Jones, (2006, Spring). Chicago Wilderness Magazine Field Notes: Telling a Tree’s Age. Retrieved from http://www.chicagowildernessmag.org/issues/spring2006/fieldnotes.html.
- Gilbert, J., Plaehn, C., (1998, March 28). Leaf Out. Retrieved from http://www.learner.org/jnorth/spring1998/critters/leaf/Update032098.html#Old.

- Tree identification guide
- Tree markers – wire clothes hangers covered with paper (from dry cleaner)
- Cross section of a tree (if available)
- String
- Markers
- Yardsticks
- Calculators
- Clipboards (optional)
- Overhead projector or interactive whiteboard
- Small whiteboard
- Whiteboard marker and eraser
- Ranger Talk or Lumberjack Talk Activity Sheet
- Ranger Table or Lumberjack Tables
- Tree Rings Overhead
- Ranger Talk or Lumberjack Talk Overhead

**Assessment Options**

- Journal Entry– Tell your friend how you would determine the age of a tree that is on your tree table.
- Go for a hunt to find the oldest tree in your schoolyard or neighborhood. Justify your information by providing your work to calculate the age. Provide a photo if possible.
- Journal Entry - How can a tree that has a longer diameter be younger than a tree with a shorter diameter? Find an example using the tree table used in your lesson.

**Extensions**

- Read about cross dating and how wooden objects can be dated based
on
*dendrochronology*, or tree-ring dating. Explain the math involved to the class through an oral presentation or Power Point presentation. - If it’s possible to visit a local park, look for a downed tree and count the rings to check its age. Also find its diameter. Find other trees in the same general area of the same type. Using what you know about the age of the downed tree, see if you can compute the age of the live trees using a ratio or proportion equation.
- Ask students to approximate the size of their tree 20 years ago. Then have them discuss or write an explanation of this African Proverb: “The best time to plant a tree is 20 years ago. The next best time is today.”
- Graph a species’ diameter versus its age using either of the Lumberjack Tables. Is it a straight line? If not, why do you think it isn’t? Compare different trees graphs to see if the slopes are different. How does the slope change if the tree is a faster growing tree?

**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 year.

[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. ]

**Teacher Reflection**

- 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?

### Learning Objectives

Students will:

- Measure circumference.
- Compute diameter when circumference is known.
- Use a table to determine best estimation of tree age.

### NCTM Standards and Expectations

- Recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life.

- Develop and use formulas to determine the circumference of circles and the area of triangles, parallelograms, trapezoids, and circles and develop strategies to find the area of more-complex shapes.

- Formulate questions, design studies, and collect data about a characteristic shared by two populations or different characteristics within one population.

- Recognize geometric ideas and relationships and apply them to other disciplines and to problems that arise in the classroom or in everyday life.

- Collect data using observations, surveys, and experiments.

- Select and apply appropriate standard units and tools to measure length, area, volume, weight, time,

### Common Core State Standards – Mathematics

Grade 3, Measurement & Data

- CCSS.Math.Content.3.MD.A.2

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

- CCSS.Math.Content.3.MD.B.4

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

- CCSS.Math.Content.4.MD.A.2

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

- CCSS.Math.Content.5.G.B.4

Classify two-dimensional figures in a hierarchy based on properties.

Grade 7, Geometry

- CCSS.Math.Content.7.G.B.4

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

- CCSS.Math.Content.7.SP.A.1

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

- CCSS.Math.Content.7.SP.A.2

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

- CCSS.Math.Content.8.SP.A.1

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.

Grade 4, Measurement & Data

- CCSS.Math.Content.4.MD.A.1

Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), ...

Grade 5, Measurement & Data

- CCSS.Math.Content.5.MD.A.1

Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

Grade 6, Stats & Probability

- CCSS.Math.Content.6.SP.A.2

Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP1

Make sense of problems and persevere in solving them.

- CCSS.Math.Practice.MP4

Model with mathematics.

- CCSS.Math.Practice.MP5

Use appropriate tools strategically.

- CCSS.Math.Practice.MP7

Look for and make use of structure.