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 Addressed

Grade 3

  • Geometry

    • The student solves problems using perimeter or area.

Grade 4

  • Geometry

    • The student solves problems using perimeter or area.

Grade 5

  • Geometry

    • The student solves problems (including real-world situations) using perimeter or area.

Grade 6

  • Geometry

    • The student solves problems (including real-world situations) using perimeter, area, or volume.

Grade 7

  • Geometry

    • The student solves problems (including real-world situations).

Grade 8

  • Geometry

    • The student solves problems (including real-world situations).

Grade 9

  • Geometry

    • The student solves problems (including real-world situations).

Grade 10

  • Geometry

    • The student solves problems (including real-world situations).

Grade 7

  • Measurement and Geometry

    • 2.0 Students compute the perimeter, area, and volume of common geometric objects and use the results to find measures of less common objects. They know how perimeter, area, and volume are affected by changes of scale

Third Grade

  • Geometry

    • Reason with shapes and their attributes.

  • Measurement and Data

    • Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

Sixth Grade

  • Geometry

    • Solve real-world and mathematical problems involving area, surface area, and volume.

Seventh Grade

  • Geometry

    • Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Grades 6-8

  • Measurement

    • Apply appropriate techniques, tools, and formulas to determine measurements

Grades 9-12

  • Geometry

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

  • Measurement

    • Apply appropriate techniques, tools, and formulas to determine measurements

Grade 6

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 2: The learner will select and use appropriate tools to measure two- and three-dimensional figures.

6th Grade

  • Measurement

    • The student will demonstrate through the mathematical processes an understanding of surface area; the perimeter and area of irregular shapes; the relationships among the circumference, diameter, and radius of a circle; the use of proportions to determine
    • The student will demonstrate through the mathematical processes an understanding of surface area; the perimeter and area of irregular shapes; the relationships among the circumference, diameter, and radius of a circle; the use of proportions to determine unit rates; and the use of scale to determine distance.

5th Grade

  • Measurement

    • The student will demonstrate through the mathematical processes an understanding of the units and systems of measurement and the application of tools and formulas to determine measurements.

7th Grade

  • Measurement

    • The student will demonstrate through the mathematical processes an understanding of how to use ratio and proportion to solve problems involving scale factors and rates and how to use one-step unit analysis to convert between and within the U.S. Customary
    • The student will demonstrate through the mathematical processes an understanding of how to use ratio and proportion to solve problems involving scale factors and rates and how to use one-step unit analysis to convert between and within the U.S. Customary System and the metric system.

8th Grade

  • Measurement

    • The student will demonstrate through the mathematical processes an understanding of the proportionality of similar figures; the necessary levels of accuracy and precision in measurement; the use of formulas to determine circumference, perimeter, area, and
    • The student will demonstrate through the mathematical processes an understanding of the proportionality of similar figures; the necessary levels of accuracy and precision in measurement; the use of formulas to determine circumference, perimeter, area, and volume; and the use of conversions within and between the U.S. Customary System and the metric system.

Geometry

  • Geometry

    • Standard G-3: The student will demonstrate through the mathematical processes an understanding of the properties and special segments of triangles and the relationships between and among triangles.

Grade 6

  • Measurement

    • 8. The student solves application problems involving estimation and measurement of length, area, time, temperature, volume, weight, and angles.

Grade 8

  • Measurement

    • 10. The student describes how changes in dimensions affect linear, area, and volume measures.

7th Grade

  • Measurement

    • 7.7 The student, given appropriate dimensions, will
    • 7.7a The student, given appropriate dimensions, will estimate and find the area of polygons by subdividing them into rectangles and right triangles

5th Grade

  • Measurement

    • 5.10 The student will differentiate between perimeter, area, and volume and identify whether the application of the concept of perimeter, area, or volume is appropriate for a given situation.
    • 5.11d The student will choose an appropriate measuring device and unit of measure to solve problems involving measurement of area - square units
    • 5.11d The student will choose an appropriate measuring device and unit of measure to solve problems involving measurement of area — square units

Textbooks Aligned

Grade Six

  • Covering and Surrounding

    • Investigation Two: Measuring Odd Shapes

  • Prime Time

    • Investigation Three: Factor Pairs

6th

  • Module 5 - Creating Things

    • Section 5: Area
      • Reason for Alignment: The Area lesson contains several worksheets guiding students through the activities listed.

7th

  • Module 6 - Flights of Fancy

    • Section 1: Inequalities and Polygons
      • Reason for Alignment: Area is an introductory lesson and contains the basics. There is a discussion of how to approach area of irregular figures which goes well with the lesson in the text of separating polygons into triangles and parallelograms to find their areas.

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 for experimenting with the activities.

Teacher Preparation

Key Terms

area

The number of square units needed to cover a surface

perimeter

The sum of the lengths of all the sides of a polygon

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 Explorer 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 Explorer 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 Outline

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.