Length, Perimeter, and Area

Abstract

This lesson is designed to examine the mathematical concepts of length, perimeter, and area. These activities and discussions may be used to develop students' understanding of these mathematical concepts.

Objectives

Upon completion of this lesson, students will:

  • be able to calculate the area and perimeter of a random shape on a grid.
  • be able to calculate the area and perimeter of a random triangle on a grid.

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;

Student Prerequisites

  • Arithmetic: Students must be able to:
    • add, count
  • Technological: Students must be able to:
    • use a calculator to square numbers
    • 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:

Key Terms

This lesson introduces students to the following terms through the included discussions:

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 trace the perimeter of a polygon that is drawn on the board, or you may begin the day by running the perimeter of the school!

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

  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 perimeter and area of polygons. We will learn exactly what these terms mean, and we will learn how to calculate area and perimeter for certain polygons.

    • We are going to use the computers to learn about area and perimeter, 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 Shape Explorer applet first.

  3. Teacher Input

    You may choose to lead the students in a short discussion about how to find length, perimeter, and area for 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 our computer applets.

    Part One: general shapes

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

    • Perimeter is the total length around the object. So imagine the grid lines are equal to one step. And imagine the outside edges of the figure are tight ropes. You want to see how many steps it will take you to get all the way around the edge. The number of steps would be the perimeter.

    • Area is the amount of space in side the figure. So imagine the grid lines mark off floor tiles like the ones we have here at school. Now floor tiles are one-foot squares. So to find the amount of space in the object we just need to count the floor tiles.

    • Once we have calculated the area and perimeter we will put our answers in the textfields and click the check answer button. If we got it right we will move on to harder shapes by using the adjust max size scroll bar.

    • If you choose to, you may pass out the Worksheet to Accompany the "Shape Explorer" Applet

    Part two: triangle area

    • Open your browser to Triangle Explorer applet in order to demonstrate this activity to the students.

    • Plot a easy triangle. Explain how the area of a triangle is 1/2 * base * height. Convince them that this is the case by clicking the hint button. They should know that the area of a rectangle is base * height and the triangle is filling half of the rectangle.

    • Next, plot a medium triangle click the hint button. Show them how the area of the medium triangles is the area of the box minus the area of the two simple triangles. Now move on up to hard triangles. Show them how the area of the triangles is the area of the box minus the area of the three simple triangles.

    • Now provoke their intrigue by asking the question "Why do we calculate the area of non-right triangles differently?" Discuss this for a few minutes. See if anyone notices that the medium type triangles follow the formula. If not introduce this by plotting a triangle and making a chalange "I will give a free one hundred to anyone who can find the area of a medium triangle faster than I can." They will not be able to beat you if you use the 1/2 * base * height formula. They will begin to ask how you do it. Show them. Plot a triangle have them figure the area. Then have them find the base and the height. Then use the formula. They will be shocked.

    • If you choose to, you may pass out the Worksheet to Accompany the "Triangle Explorer" Applet

  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 another example together, but let the students direct your actions:

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

      • Can someone describe how I would find the area of this 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 to be sure that 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 and perimeter using the Shape Explorer applet. Who ever 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. Switch to the Triangle Explorer applet and do the same.

  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:

    • Don't play the game for speed. 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 and the perimeter (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. Switch to the Triangle Explorer applet and do the same.

    Extensions

    This lesson may be followed by: