Students will estimate the surface of the pond without using formulas. Students explore the use of chance methods for estimating surface area.
Hole punch; Blank sheet of paper; Graph paper; Markers; Blindfold; photocopied map of pond on grid paper. (If you can't find a map of your pond, you may use a printout of the virtual pond.);
Background For Teachers:
In this activity the students will estimate the surface area of a pond using two estimation methods. The chance method requires a student to 'randomly' drop small paper circles (created with a hole punch) onto the map of the pond. Students then count the number of paper circles touching the pond water grid squares and then calculate an estimated surface area for the pond. The second method requires the students to average a maximum pond area estimate and a minimum pond area estimate.
Intended Learning Outcomes:
Students will explore the use of a chance method to estimate the surface area of a pond.
Ask the students to imagine that they have fallen from an orbiting space station and somehow survive the fall through the atmosphere. Where are you most likely to land, in the water or on dry ground? Since there is more surface area covered by water than by land, you are more likely to land in the water. Mathematicians use the fact that randomly falling objects are more likely to hit objects with large surface areas than they are to hit objects with small surface areas to estimate surface areas. Divide the students into groups of at least three members. Explain that the groups will estimate the surface area of a pond using two estimation methods. Chance Method - Step 1 - One group member creates ten small paper circles using the hole punch and the blank sheet of paper. A second group member may need to 'catch' the paper circles as they fall out of the hole punch. The paper circles need not be perfect circles, but they should be about the same size. A third group member should randomly drop (closed eyes or blinfold will increase randomness here) the ten paper circles onto the map of the pond. Chance Method - Step 2 - The group should count the number of paper circles that fell onto a map grid representing pond water. On the virtual pond map found at www.uen.org/pond/ the pond water grids are in light blue. The group should also count the total number of grid squares on the map (There are about 850 complete grid squares). Chance Method - Step 3 - The group must now calculate the estimated surface area from the results. (As an example assume that 3 of the 10 circles landed on a pond water grid square. The ratio would be 3 pond water circles to 10 total circles. Using this ratio the group can estimate the total number of pond water grid squares among the 850 total grid squares (3 is to 10 as 30 is to 100 as 240 is to 800. Add in 15 for the additional half a hundred grid squares and we would estimate about 255 pond water grid squares for this example).The teacher may need to help the students discover the ratio of the total by stepping them through as shown in the example above. Averaging Max and Min Estimations Method - Step 1 - The group carefully looks at the pond map and counts the number of grid squares that contain ANY pond water. Teacher's Note: Determining if a grid square contains any water is a very difficult task and most students will have some trouble deciding whether or not to count a grid square. On the map of the virtual pond provided at www.uen.org/pond/ there is a row with the word 'level' appearing in it's left side. There would be three grid squares that contain some water but only 1 grid square that contains ONLY water. Be flexible with the student counts but there are about 110 squares that contain some water. Averaging Max and Min Estimation Method - Step 2 - The group should count the total number of grid squares that contain only water. Teacher Note: On the map of the virtual classroom found at www.uen.org/pond/ there are about 67 such grid squares that contain only water. Averaging Max and Min Estimation Method - Step 3 - The group should now average the number of grid squares containing some water and the number of grid squares containing only water ((110 + 67)/2 = 88.5) to obtain an estimate of the surface area of the pond. Comparing the Two Methods - Step 1 - Have students create a bar graph showing the total number of pond water squares estimated by the chance method. On the same graph show a bar representing the total number of water squares estimated by the averaging method. Comparing the Two Methods - Step 2 - Assign the groups to write a short report comparing the two results. In the report the group should discuss what factors may have affected the results of each method. The report may also discuss under what situations each method would be the most reasonable method to use.
This might be a good time to discuss the concepts of 'fair' and 'biased'. When the counties of Utah were first created each county met the criteria for a minimum number of residents. In many organizations, each county gets an equal vote. Since the populations have changed over the years, what biases might this type of voting encourage? In the United States Senate, every state receives an equal number of senators. Explore how the Utah Senate is organized. Do counties get an equal number of senators? Why do you think Utah State Senators are selected based on geography and population?
Completion of a graph and short report comparing the results of the two methods.
Created Date :
Mar 19 1999 13:38 PM