Skip to main content Skip to main content
Search
Close this search box.
Search
Close this search box.

Private: Learning Math: Measurement

Professional Development > Private: Learning Math: Measurement > 6. Area > 6.4 Homework

Mathematics

K-2, 3-5, 6-8

Area Homework

Session 6, Homework

Problem H1

If the area of each of the smallest two triangles in the tangram below is equal to 1 unit, find the area of each of the other pieces, and then find the area of the entire tangram:

 

Problem H2

The formula for the area of a trapezoid is A = , where b1 and b2 represent the top and bottom base of the trapezoid and h represents the height. Draw a trapezoid on a sheet of paper, and connect either of the two opposite vertices. Into what shapes has the trapezoid been divided? What are the height and base of each shape? Find the area of each shape and add them together. How does this area compare to the total area of the trapezoid?

 

Problem H3

Examine the trapezoid below:

Find the area of this trapezoid using the formula A = . Then find the area in a different way. How do the two areas compare?

 

Take It Further:

Problem H4

It is possible to find a formula for the area of geoboard polygons as a function of boundary dots and interior dots. For example, the two polygons below each have five boundary dots and three interior dots:

  1. What is the area of each polygon?Let’s gather data to help us find what’s known as Pick’s formula, which is used for determining the area of a simple closed curve (in our case, the areas of the polygons on a geoboard).For Problems (b)-(d), build figures on the geoboard or draw figures on dot paper that have the indicated number of boundary dots (b) and interior dots (I). Find the area of each figure, and record your results in the tables below.
  2. If I = 0, calculate the area of each figure:

    Number of Boundary Dots

    Area (in Square Units)
    3
    4
    5
    6
    7
    b
  3. If I = 1, calculate the area of each figure:

    Number of Boundary Dots

    Area (in Square Units)
    3
    4
    5
    6
    7
    b
  4. If I = 2, calculate the area of each figure:

    Number of Boundary Dots

    Area (in Square Units)
    3
    4
    5
    6
    7
    b
  5. What patterns do you notice in these tables? Each time you add a boundary dot, how does it change the area?
  6. Find a formula for the area of a geoboard figure if it has b boundary dots and I interior dots.

Suggested Reading

Fan, C. Kenneth (January, 1997). Areas and Brownies. Mathematics Teaching in the Middle School, 2 (3), 148-160.
Reproduced with permission from Mathematics Teaching in the Middle School. © 1997 by the National Council for Teachers of Mathematics. All rights reserved.

Download PDF File:
Areas and Brownies

Solutions

Problem H1

Using the fact that each of the tangram pieces can be divided into some number of small triangles, we get the following areas: The small square, the parallelogram, and the dmedium triangle each have areas of 2 square units, and the large triangles each have areas of 4 square units. The area of the entire tangram is 16 square units.

 

Problem H2

The trapezoid is divided into two triangles, each with height h, the height of the trapezoid. One triangle has base b1 while the other has base b2:

Since the area of a triangle is , the total area is , or , which is also the area of the trapezoid.

 

Problem H3

Using the Pythagorean theorem, a2 + b2 = c2, we can calculate the lengths of the bases and the height. The bases have lengths of 2√2 (the hypotenuse of the triangle with legs 2 and 2) and 6√2 (the hypotenuse of the triangle with legs 6 and 6), respectively. Similarly, the height is 2√2 (it is the length perpendicular to the bases and the hypotenuse of the triangle with legs 2 and 2). The area is , which equals 16 square units.

We can also find the area by drawing a rectangle around the trapezoid and then subtracting the smaller areas, or by dividing the trapezoid into two triangles, or by any of several other methods. But any way you slice it, the area is 16 square units.

To learn more about the Pythagorean theorem, go to Learning Math: Geometry, Session 6.

 

Problem H4

    1. Both shapes have areas of 4.5 square units.

    2. Number of Boundary Dots

      Area
      (in Square Units)
      3 0.5
      4 1
      5 1.5
      6 2
      7 2.5
      b
      – 1

      In each case, the area is half the number of boundary dots minus 1. If there are b boundary dots and one interior dot, the area is  -1.

    3. Number of Boundary Dots

      Area
      (in Square Units)
      3 1.5
      4 2
      5 2.5
      6 3
      7 3.5
      b

      In each case, the area is half the number of boundary dots. If there are b boundary dots and one interior dot, the area is .

    4. Number of Boundary Dots

      Area
      (in Square Units)
      3 2.5
      4 3
      5 3.5
      6 4
      7 4.5
      b
      + 1
    5. In each case, the area is half the number of boundary dots plus 1.
      If there are b boundary dots and two interior dots, the area is + 1.
      Every time you add a boundary dot, the area goes up by half a unit, regardless of the orientation of the shape.
    6. The formula is A = I + – 1, and it can be used on many of the figures in Part B of this session.

     

Series Directory

Private: Learning Math: Measurement

Sections

6.1 Part A: Measuring Area (35 minutes)

Session 6, Part A

6.2 Part B: Exploring Area With a Geoboard (50 minutes)

Session 6, Part B

6.3 Part C: Scaling the Area (35 minutes)

Session 6, Part C

6.4 Homework

Session 6, Homework

Sessions

How to Use 0 About the Learning Math Courses

Session 1 What Does It Mean To Measure?

Explore what can be measured and what it means to measure. Identify measurable properties such as weight, surface area, and volume, and discuss which metric units are more appropriate for measuring these properties. Refine your use of precision instruments, and learn about alternate methods such as displacement. Explore approximation techniques, and reason about how to make better approximations.

Session 2 Fundamentals of Measurement

Investigate the difference between a count and a measure, and examine essential ideas such as unit iteration, partitioning, and the compensatory principle. Learn about the many uses of ratio in measurement and how scale models help us understand relative sizes. Investigate the constant of proportionality in isosceles right triangles, and learn about precision and accuracy in measurement.

Session 3 The Metric System

Learn about the relationships between units in the metric system and how to represent quantities using different units. Estimate and measure quantities of length, mass, and capacity, and solve measurement problems.

Session 4 Angle Measurement

Review appropriate notation for angle measurement, and describe angles in terms of the amount of turn. Use reasoning to determine the measures of angles in polygons based on the idea that there are 360 degrees in a complete turn. Learn about the relationships among angles within shapes, and generalize a formula for finding the sum of the angles in any n-gon. Use activities based on GeoLogo to explore the differences among interior, exterior, and central angles.

Session 5 Indirect Measurement and Trigonometry

Learn how to use the concept of similarity to measure distance indirectly, using methods involving similar triangles, shadows, and transits. Apply basic right-angle trigonometry to learn about the relationships among steepness, angle of elevation, and height-to-distance ratio. Use trigonometric ratios to solve problems involving right triangles.

Session 6 Area

Learn that area is a measure of how much surface is covered. Explore the relationship between the size of the unit used and the resulting measurement. Find the area of irregular shapes by counting squares or subdividing the figure into sections. Learn how to approximate the area more accurately by using smaller and smaller units. Relate this counting approach to the standard area formulas for triangles, trapezoids, and parallelograms.

Session 7 Circles and Pi (π)

Investigate the circumference and area of a circle. Examine what underlies the formulas for these measures, and learn how the features of the irrational number pi (π) affect both of these measures.

Session 8 Volume

Explore several methods for finding the volume of objects, using both standard cubic units and non-standard measures. Explore how volume formulas for solid objects such as spheres, cylinders, and cones are derived and related.

Session 9 Measurement Relationships

Examine the relationships between area and perimeter when one measure is fixed. Determine which shapes maximize area while minimizing perimeter, and vice versa. Explore the proportional relationship between surface area and volume. Construct open-box containers, and use graphs to approximate the dimensions of the resulting rectangular prism that holds the maximum volume.

Session 10 Classroom Case Studies, K-2

Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for K-2 students.

Session 11 Classroom Case Studies, 3-5

Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 3-5 students.

Session 12 Classroom Case Studies, 6-8

Watch this program in the 10th session for grade 6-8 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 6-8 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 6-8 students.

Join us for conversations that inspire, recognize, and encourage innovation and best practices in the education profession.

Available on Apple Podcasts, Spotify, Google Podcasts, and more.

Listen Now
Subscribe Me
Facebook Twitter Youtube
logo-white
2000 Avenue of the Stars, Suite 1000S, Los Angeles, CA 90067
2024 Annenberg Foundation. All rights Reserved.
Privacy Policy.