General Information
Subject Area: Mathematics (B.E.S.T.)
Grade: 912
Strand: Geometric Reasoning
Date Adopted or Revised: 08/20
Status: State Board Approved
Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Area
- Perimeter
- Polygon
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
In elementary grades, students are introduced to the concepts of perimeter and area, with a focus on rectangles. In middle grades, students expand their knowledge of areas of quadrilaterals and triangles. In Geometry, students find perimeter and area of two-dimensional polygons on the coordinate plane. In later courses, students develop further tools for finding area, especially using calculus.- Instruction includes discussing the convenience of answering with exact values (e.g., the simplest radical form) or with approximations (e.g., rounding to the nearest tenth or hundredth). It is also important to explore the consequences of rounding partial answers on the accuracy or precision of the final answer, especially when working in real-world contexts.
- Problem types include polygons that are convex (where all interior angle measures are less than 180 degrees), concave (where at least one interior angle measure is more than 180 degrees), regular polygons (where all interior angle measures and side lengths are equivalent) and irregular polygons.
- Instruction includes the use of the distance formula and the Pythagorean Theorem to find side lengths of the polygon in order to determine the perimeter or area.
- Instruction includes various methods to determine area on the coordinate plane.
- Decomposition
- Students can decompose the polygon into triangles and rectangles (or other quadrilaterals) to determine the total area by adding each of the partial areas.
- Subtraction Method (Box Method)
- Students can draw a rectangle that includes as many vertices of the polygon as possible and then to determine the area of the polygon, subtract the area(s) of the shape(s) that is (are) in the “box” but not in the polygon from the area of the rectangle.
- For possible enrichment, other methods include, but are not limited to, Pick’s Theorem or the Shoelace method.
- Decomposition
- Refer to Appendix E for formulas of two-dimensional figures. It may be helpful to review the derivation of these formulas, as this was done in middle grades.
Common Misconceptions or Errors
- Students may confuse the concepts of area and perimeter.
- Students may assume that the larger the area a polygon has, the larger its perimeter will
be.
- For example, a rectangle may have an area of 64 square meters and could have a perimeter of 32 meters or any perimeter larger than 32 meters.
Instructional Tasks
Instructional Task 1 (MTR.3.1, MTR.6.1)- Given parallelogram EFGH with vertices E(−1, 5), F(2, 8), G(4, 4) and H(1, 1).
- Part A. Find the exact perimeter and area of the parallelogram.
- Part B. Find the perimeter and area of the parallelogram to the nearest tenth.
- Joe’s commute to work can be represented in the coordinate plane as follows:
- His house is at H(0,0).
- His favorite coffee shop is at C(7, 6) where he stops every morning.
- His office is at W(4, 13).
- He goes back home from his office every day without stopping.
- Part A. Assume that Joe lives in a city where the roads are parallel to the coordinate axes and each intersection occurs at integer coordinates. Represent his route on the coordinate plane where each city block is one coordinate unit by one coordinate unit, which measures 175 yards by 175 yards.
- Part B. What is the total distance, in yards, that Joe commutes every day, assuming that he stays on the roads?
- Part C. If Joe could take the most direct route (cutting across city blocks) for his commute, what would be his total distance, in yards, that he commutes every day?
Instructional Items
Instructional Item 1- The Move With Us Run Team is planning a run around the combined perimeter of Polk and
Osceola counties (as shown by the green rectangle).
- Part A. What are the coordinates of the four vertices that could be used to measure the run around the two counties? Use the scale provided on the map to determine the coordinates.
- Part B. Using the coordinates found in Part A, what would be the total distance of the run, in miles?
- Part C. Assume that the group runs a total of 10 miles every day, how many days would it take them to complete the distance around the two counties?