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 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).

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.
    • Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.

Fifth Grade

  • Geometry

    • Classify two-dimensional figures into categories based on their properties.

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 3-5

  • Measurement

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

Grades 6-8

  • 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.

Technical Mathematics I

  • Geometry and Measurement

    • Competency Goal 2: The learner will measure and apply geometric concepts to solve problems.

Technical Mathematics II

  • Geometry and Measurement

    • Competency Goal 1: The learner will use properties of geometric figures to solve problems.

3rd Grade

  • Measurement

    • The student will demonstrate through the mathematical processes an understanding of length, time, weight, and liquid volume measurements; the relationships between systems of measure; accurate, efficient, and generalizable methods of determining the perim
    • The student will demonstrate through the mathematical processes an understanding of length, time, weight, and liquid volume measurements; the relationships between systems of measure; accurate, efficient, and generalizable methods of determining the perimeters of polygons; and the values and combinations of coins required to make change.

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.

4th Grade

  • Measurement

    • Standard 4-5: The student will demonstrate through the mathematical processes an understanding of elapsed time; conversions within the U.S. Customary System; and accurate, efficient, and generalizable methods of determining area.

7th Grade

  • Geometry

    • The student will demonstrate through the mathematical processes an understanding of proportional reasoning, tessellations, the use of geometric properties to make deductive arguments. the results of the intersection of geometric shapes in a plane, and the
    • The student will demonstrate through the mathematical processes an understanding of proportional reasoning, tessellations, the use of geometric properties to make deductive arguments. the results of the intersection of geometric shapes in a plane, and the relationships among angles formed when a transversal intersects two parallel lines.

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-1: The student will understand and utilize the mathematical processes of problem solving, reasoning and proof, communication, connections, and representation.
    • Standard G-2: The student will demonstrate through the mathematical processes an understanding of the properties of basic geometric figures and the relationships between and among them.
    • 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.

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

4th Grade

  • Measurement

    • 4.13.b
    • 4.13.b The student will use measuring devices to find perimeter in both standard and nonstandard units of measure.

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.
    • 5.11a The student will choose an appropriate measuring device and unit of measure to solve problems involving measurement of length - part of an inch (1/2, 1/4, and 1/8), inches, feet, yards, miles, millimeters, centimeters, meters, and kilometers
    • 5.11a The student will choose an appropriate measuring device and unit of measure to solve problems involving measurement of length — part of an inch (1/2, 1/4, and 1/8), inches, feet, yards, miles, millimeters, centimeters, meters, and kilometers

8th Grade

  • Computation and Estimation

    • 8.3 The student will solve practical problems involving rational numbers, percents, ratios, and proportions. Problems will be of varying complexities and will involve real-life data, such as finding a discount and discount prices and balancing a checkbook.
    • 8.5 The student, given a whole number from 0 to 100, will identify it as a perfect square or find the two consecutive whole numbers between which the square root lies.
    • 8.3 The student will solve practical problems involving rational numbers, percents, ratios, and proportions. Problems will be of varying complexities and will involve real-life data,
    • 8.5 The student, given a whole number from 0 to 100, will identify it as a perfect square or

  • Geometry

    • 8.10a The student will verify the Pythagorean Theorem, using diagrams, concrete materials, and measurement; and
    • 8.10 The student will

  • Measurement

    • 8.6 The student will verify by measuring and describe the relationships among vertical angles, supplementary angles, and complementary angles and will measure and draw angles of less than 360°.
    • 8.6 The student will verify by measuring and describe the relationships among vertical

  • Patterns, Functions, and Algebra

    • 8.17 The student will create and solve problems, using proportions, formulas, and functions.

6th Grade

  • Measurement

    • 6.11 The student will determine if a problem situation involving polygons of four or fewer sides represents the application of perimeter or area and apply the appropriate formula.

Textbooks Aligned

Grade Six

  • Covering and Surrounding

    • Investigation One: Measuring Perimeter and Area
    • Investigation Three: Constant Area, Changing Perimeter
    • Investigation Four: Constant Perimeter, Changing Area
    • Investigation Six: Measuring Triangles

  • Shapes and Designs

    • Investigation Two: Building Polygons

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 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 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 inside 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 the Triangle Explorer applet in order to demonstrate this activity to the students.
    • Plot an 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 challenge "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.
    • 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 Outline

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.