Learning Math: Measurement
Area Part A: Measuring Area (35 minutes)
Session 6, Part A
In This Part
- Measuring a Surface
- Areas of Irregular Shapes
Measuring a Surface
Think of some situations that involve the measurement of area. How would you determine these areas? For example, in order to determine how much paint to buy for your living room, how would you find the area of each surface you plan to paint — your walls, ceiling, etc.?
Units of measurement for area involve shapes that cover a plane. Some shapes, such as rectangles and squares, cover the plane more completely than other shapes, such as circles. As mentioned above, squares are the most common unit of measurement in the U.S. customary system (square inches, square feet, square yards, square miles). The standard unit in the metric system is the square meter, with square centimeters and hectares (equivalent to 10,000 square meters) used for smaller and larger surfaces.
Most people are familiar with methods for finding the area of familiar shapes, such as rectangles and squares. But is it possible to determine the area of an irregularly shaped object? How would you determine such an area without a formula? Let’s explore.
Areas of Irregular Shapes
Trace your hand on a piece of paper. Think about how you might determine the area of your handprint.
Will the amount of area covered differ if you trace your hand with your fingers close together or spread apart? Explain.
Think about the activity you did with the tangram triangles in Session 1, Part C.
Units for measuring area must have the following properties:
- The unit itself must be the interior of a simple closed shape.
- The unit, when repeated, must completely cover the object of interest, with no holes or gaps (like a tessellation).
- Many polygons (e.g., rectangles, rhombuses, and trapezoids) and irregular shapes (e.g., L shapes) have this property and can thus be used as units of measurement.
- What units might you use to determine the area of your handprint?
- Why can’t you use a small circle as the unit of measurement?
- One method for finding the area of an irregular shape is to count unit squares. Use centimeter grid paper (PDF – be sure to print this document full scale) to determine the area of your handprint. What are the disadvantages of this method?
- Another method is to subdivide your handprint into sections for which you can easily calculate the area. Find the area of your handprint using this method. Does using the two methods result in the same area?
Up until now, you have been approximating the area of your handprint. In other words, your measurements were not exact.
What can you do to make your approximation more accurate? Explain why this approach will lead to a better approximation.
Another way to approximate the area of a handprint or any other irregular shape is to determine the number of squares that are completely covered and the number of squares that are partially covered. Average these two numbers to get an approximate area in the number of square units.
Think about the following statement: If you repeatedly use a smaller and smaller unit to calculate the area of an irregular shape, you will get a closer and closer approximation and eventually find the exact area. What do you think of this line of reasoning? Explain.
The palm of your hand is about one percent of your body’s surface area. Doctors sometimes use this piece of information to estimate the percent of the body that is affected in burn victims. Use your data to approximate the amount of skin on your body.
No, the surface area should be identical, since it is the fingers, not their location, that determine the area. The same shapes in different locations will have the same area.
- A good unit of measurement might be a square centimeter or a square millimeter on grid paper. An important aspect is that the unit should tile well.
- A circle does not tile well; circles leave holes in between them, and counting the area of the circles would not approximate the total area of the hand.
- Answers will vary. Counting squares is subject to several errors, including the possibility of miscounting the squares and the rounding error introduced by trying to count “half squares.” Additionally, it may not be the most precise method. It also takes a while to do.
- Answers will vary. Using two different methods will most likely result in different numerical values, since the area in both cases is an approximation and therefore subject to error.
Answers will vary. One useful method is to use paper with a finer grid, as smaller squares (units) will result in fewer rounding errors.
This is a reasonable method of approximation, although there are still other forms of measurement error that can have an effect on the calculation. We can increase the accuracy of our measurement by making the units smaller, for example, using mm2instead of cm2 grid paper. But because we are physically measuring it, the area will always be an approximation, no matter how small the unit (and there’s always a smaller unit!). The measurement process always results in an approximate rather than an exact value.
Answers will vary. You can make this approximation by multiplying your palm’s area by 100.
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.