Prerequisite skills:

Geometry and Algebra
II Knowledge of the “Law of Sines” for computing lengths and area

NCTM (PA) Math Standards:

• 2.2.11.D -- Describe and explain the amount of error that may exist in a computation using estimates.
• 2.2.11.E -- Recognize that the degree of precision needed in calculating a number depends on how the results will be used and the instruments used to generate the measure.
• 2.3.11.B -- Measure and compare angles in degrees and radians.
• 2.3.11.C -- Demonstrate the ability to produce measures with specified levels of precision.
• 2.4.11.E -- Demonstrate mathematical solutions to problems (e.g., in the physical sciences).
• 2.5.11. A -- Select and use appropriate mathematical concepts and techniques from different areas of mathematics and apply them to solving non-routine and multi-step problems.
• 2.5.11.B -- Use symbols, mathematical terminology, standard notation, mathematical rules, graphing and other types of mathematical representations to communicate observations, predictions, concepts, procedures, generalizations, ideas and results.
• 2.5.11.C -- Present mathematical procedures and results clearly, systematically, succinctly and correctly.
• 2.5.11.D -- Conclude a solution process with a summary of results and evaluate the degree to which the results obtained represent an acceptable response to the initial problem and why the reasoning is valid.
• 2.7.11.B -- Apply probability and statistics to perform an experiment involving a sample and generalize its results to the entire population.
• 2.9.11.I -- Model situations geometrically to formulate and solve problems.

Lab Objectives:

• Students will measure angles of triangles with protractors.
• Students will calculate area of triangles using the “Law of Sines”, “Heron’s Formula”, “Divide and Conquer” and “Pick’s Method”.
• Students will construct triangles from irregular shapes on Square Dot Paper.
• Students will use the “Monte Carlo” procedure to predict area via probability.
• Students will analyze solutions from the different procedures for calculating area to decide the accuracy of their results.

Materials:

• Handouts (located at the bottom of this lab)
• Lab Sample Hand-out
• Student Project problem Hand-out
• Protractor
• Graphics Calculators
• Geometric Sketchpad (to make more irregular shapes)

Area of Irregular shapes using Triangles:

Sample and Procedures:
Observe the grid below (each grid square represents 1 square unit):

Project Objective:
Find the area of the irregular portion of the grid (lower left portion) using only triangular portions. Student pairs will compute the area via counting grid points and measuring angles (with a protractor) only. Students will not rely on a ruler to measure sides. Use the following methods to find the area of the figure:

• Divide and Conquer
• “Law of Sines”
• Heron’s Formula
• Pick’s Method
• Monte Carlo

Procedure:

Count grid marks to find the area of the rectangle. For this example, the length measures 18 units and the width measures 10 units. The rectangle has an area of 180 square units. This information will be used with the Monte Carlo method.

“Divide and Conquer”:

1. Break up the figure using the Divide and Conquer Method of approximating square units. In this method, try to break the object up into squares and right triangles. This is the only option that will allow you to use non-triangular objects. Observe one way to break up the irregular shape using “Divide and Conquer”:



2. Find the areas of each figure you drew and calculate the area of the irregular shape.
   



3. Use “Divide and Conquer” again but only break the irregular shape up into right triangles. Observe one possibility (the figure below was reduced to save space):

   

   Find the Area of all four Right Triangles:

   

“Use “Law of Sines” Area Formula”:

1. Break the irregular shape up into two triangles only. Make sure you find a pair of triangles so that you can count the unit measure of one side.

   

2. Observe the “Law of Sines” area formula:
   
3. Use your protractor to measure the angles of both triangles.
   
4. Use the “Law of Sines” and/or count obvious units to find the measure of at least two sides of each triangle. Use these results to calculate area:
   
   Solutions:
   

“Heron’s Formula”:

1. Since you already found two sides for each triangle in the previous exercise, all you need to do first is find the final sides of each triangle using “Law of Sines”.
2. Solve each triangle using Heron’s Formula and sum the results:
   

Pick’s Formula:
Pick’s formula uses boundary and interior points as follows:

     *b --> number of points on the boundary of the region
     *c --> number of points in the interior of the region
There are 15 points on the boundary and 26 points in the interior.

Solution: