These model lessons were created by teachers participating in the Minnesota Department of Education's 2011-13 project, Integrating Environmental and Outdoor Education into Grades 7-12 with funding from the Minnesota Environment and Natural Resources Trust Fund as recommended by the Legislative-Citizen Commission on Minnesota Resources.
Title of lesson: Seeing The Woods from the Trees
Content area: Mathematics/Science
Grade level: 7
Students will be able to calculate the diameter, radius, and approximate height and the approximate density of trees in the Camden Neighborhood of North Minneapolis using perpendicular angles and formulas for a circle, area, volume, and density.
Standards or benchmarks addressed:
MN Math Standards-2007
184.108.40.206 - Understand calculators and other computing technologies often truncate or round numbers.
220.127.116.11 - Use proportional reasoning to solve problems involving ratios in various contexts
18.104.22.168 - Understand that a relationship between two variables, x and y, is proportional if it can be expressed in the form y/x = k or y = kx. Distinguish proportional relationships from other relationships, including inversely proportional relationships.
22.214.171.124 - Solve multi-step problems involving proportional relationships in numerous contexts.
126.96.36.199 - Use knowledge of proportions to assess the reasonableness of solutions.
188.8.131.52 - Represent relationships in various contexts with equations involving variables and positive and negative rational numbers. Use the properties of equality to solve for the value of a variable. Interpret the solution in the original context.
184.108.40.206 - Demonstrate an understanding of the proportional relationship between the diameter and circumference of a circle and that the unit rate (constant of proportionality) is. Calculate the circumference and area of circles and sectors of circles to solve problems in various contexts.
220.127.116.11 - Apply scale factors, length ratios and area ratios to determine side lengths and areas of similar geometric figures.
MN Science Standards-2009
18.104.22.168.1 - Explain density, dissolving, compression, diffusion and thermal expansion using the particle model of matter.
22.214.171.124.3 - Recognize that variation exists in every population and describe how a variation can help or hinder an organism's ability to survive.
126.96.36.199.2 - Determine and use appropriate safety procedures, tools, measurements, graphs and mathematical analysis to describe and investigate natural and designed systems in a life science context.
188.8.131.52.2 - Use physical properties to distinguish between metals and non-metals.
184.108.40.206.2 - Determine and use appropriate safety procedures, tools, measurements, graphs and mathematical analyses to describe and investigate natural and designed systems in Earth and physical science contexts.
Description of lesson and how it is adapted for EOE:
Marking five points along a line on the string
Laying another string perpendicular to the main string at each point to make four quadrants
Find the closest tree to the point measuring 4 inches across at 4 feet from the ground
Determine the width of the tree by dividing circumference by pi (3.14) or using calipers
Measure the distance in meters from the starting point to this tree and write it down
Repeat this method with three other trees in the other quarters for that point
Repeat again for the points created along the same line
Add together the distances from the four trees to the point and divide the distances by four to find the average distance of the trees to each point
Repeat this procedure for the remaining points
Add these average distances for all five points and divide by five to find the overall distance of the trees in meters
Multiply the average distance in meters by itself to find the average area each tree occupies
Divide 10,000 meters squared by the average tree area to determine the tree density per hectare
Determine the height of each tree using its shadow by indirect measurement
Calculate the average height of the trees times the average diameter and multiple this by the density to find the approximate total space density occupied by trees in the cubic space
(optional) Draw a scale model of the area and compare/contrast it with the entire 1 or 10 kilometer satellite image using Google maps per urban tree density
Students work in groups or 3-4 to measure the width, area density, and heights of four trees. To do this, students need to know the proportional relationships between circumference and diameter or how to use a caliper, and measure the length and width of an area being measured by:
The teacher models measuring techniques to the students. The teacher assigns students to cooperative learning groups with the following roles and responsibilities:
Leader/Facilitator and Note-Taker -- Is responsible for making sure every student does their part, laying the string around the area being measured for density, and for recording accurate measurements in the notebook.
Measurer -Is responsible for measuring the circumference, width (if using calipers), and length of shadows as well as measuring the string for the area being measured.
Calculator-Is responsible for calculating the width, distance, and height of trees using indirect measurement and expressing the values in meters and hectare.
Scale Model Drawer/Organizer (optional role -- see extensions)-Draws scale models of the outside and perceived inside of the trees being measured, shadows, as well as all large items in the four-quadrant area being measured.
Other resources needed:
- Outdoor Notebooks and Sharpened Pencils
- Metric Measuring Tapes
- Calipers (optional)
- Calculator (optional)
- Google Maps (optional)
How students are assessed:
The teacher has a selected tree with fairly easy measurements that has been pre-calculated for an authentic assessment of student skill in finding the diameter, radius, and approximate density. Additional practice problems, quiz and unit assessment questions are given throughout the unit to asses understanding of math and science standards using the same or similar contextualized problems.
Demonstration of measuring techniques and using indirect measurement can be done before going outside and even the day before to build background. Students should have had prior knowledge of circle and density formulas before this activity. By using Calipers and Calculators the activity can be completed in a 50 minute session when students work in structured groups of three or four with clear roles and understanding of procedures. Having formulas readily available solved for different variables in student journals is an adaption that also increases student productivity time. In order for all students to complete the assessment additional time may be needed as well.
Environmental and Outdoor Education (EOE) Model Lessons are freely available for use by all teachers for educational purposes only.