Area

Abstract

This lesson is designed to examine the mathematical concept of area within fixed perimeters. These activities and discussions may be used to develop students' understanding of this mathematical concept.

Objectives

Upon completion of this lesson, students will:

  • be able to calculate the area of a random shape on a grid.
  • be able to explain the correlation between the size of the perimeter and the number of different possible areas that can be contained within that perimeter.
  • be able to explain the affect the shape of the perimeter has on the area of the object.

Standards

The activities and discussions in this lesson address the following NCTM standards:

Geometry

Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationship

  • precisely describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties;
  • understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects;

Measurements

Apply appropriate techniques tools, and formulas to determine measurements

  • select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision;

Key Terms

This lesson introduces students to the following term(s) through the included discussions:

Activities

This lesson introduces students to the concept of area through the following activity:

Student Prerequisites

  • Arithmetic: Students must be able to:
    • add, subtract, multiply, count
    • understand perimeter (or have it included in this lesson)
  • Technological: Students must be able to:
    • perform basic mouse manipulations such as point, click and drag.
    • use a browser, such as Netscape, for experimenting with the activities.

Teacher Preparation

Students will need:

Lesson Outline

  1. Focus and Review

    Remind students what has been learned in previous lessons that will be pertinent to this lesson and/or have them begin to think about the words and ideas of this lesson:

    • Ask students to recall information about polygons.

    • You might ask students to consider how they might find the area of their desk.

    • Discuss what it might mean to talk about the area of a polygon.

    • Ask students if they know what perimeter is. If they do not understand perimeter, give a brief explanation of it.

  2. Objectives

    Let the students know what it is they will be doing and learning today. Say something like this:

    • Today, class, we will be talking more about the area of polygons. We will learn exactly what the term area means and how to calculate area for random shapes with a fixed perimeter. We will also learn what effect the shape of the perimeter has on the area of the object. Finally, we will learn how the size of the perimeter affects the number of different possible areas that can be contained by the given perimeter.

    • We are going to use the computers to learn about area, but please do not turn your computers on or go to this page until I ask you to. I want to show you a little about the Area applet first.

  3. Teacher Input

    You may choose to lead the students in a short discussion about how to find the area and perimeter of irregular figures.

    Explain to the students how to do the assignment. You should model or demonstrate it for the students, especially if they are not familiar with how to use the computer applets.

    • Open your browser to Area in order to demonstrate this activity to the students.

    • Area is the amount of space inside the figure. So imagine the grid lines mark off floor tiles like the ones we have here at school. Many floor tiles are one-foot squares. So to find the amount of space in the object we just need to count the floor tiles. That would give us the area of the floor in square feet. Similarly, we will do the same with finding the area of the objects on the grid, just count the number of squares. Note, though there is no specified unit of measure.

    • Once we have calculated the area we will put our answer in the text field and click the check answer button.

    • If you choose to, you may pass out the Worksheet to accompany the "Area" Applet and the Record sheet so that students can record the necessary information to complete the worksheet.

  4. Guided Practice

    Try another example, letting the students direct your moves. Or, you may simply ask, "Can anyone describe the steps you will take for this assignment?"

    • If your class seems to understand the process for doing this assignment, simply ask, "Can anyone tell me what you will do now?"

    • If your class seems to be having a little trouble with this process, do example together, but let the students direct your actions:

      • Can someone describe how I would find the area of this shape?

      • How does the shape of the perimeter affect the area of the shape?

  5. Independent Practice

    • Allow the students to work on their own and to complete the worksheet, should you choose to provide one. Monitor the room for questions and be sure the students are on the correct web site.

    • Another option for independent practice is to have the students work in pairs (carefully chosen so that both students are of the same ability group). Have them race to find the correct area using the Area applet. Whoever wins gets a point. At the end of the allotted time for the game give the winning member of each pair a reward of some type.

  6. Closure

    You may wish to bring the class back together for a discussion of the findings. Once the students have been allowed to share what they found, summarize the results of the lesson.

Alternate Outlines

This lesson can be rearranged in several ways if there is only one available computer:

  • Have all students pull out a sheet of paper. Have the computer generate a set number of shapes and have them record on their paper the area (you record it as well). When you are done, take up the papers and check them. The person with the most correct answers gets a reward of some type and the rest of the class gets a participation grade. That way everyone tries.

Extensions

This lesson may be followed by: