Learning Math: Measurement
What Does It Mean To Measure? Part D: Summing It Up (10 minutes)
Session 1, Part D
What is measurement?
Measurement is the process of quantifying the properties of an object by expressing them in terms of a standard unit. Measurements are made to answer such questions as, How heavy is my parcel? How tall is my daughter? How much chlorine is in this water?
How do we measure?
The process of measuring consists of three main steps. First, you need to select an attribute of the thing you wish to measure. Second, you need to choose an appropriate unit of measurement for that attribute. Third, you need to determine the number of units.
What procedures are used to determine the number of units?
Some measurements require only simple procedures and little equipment — measuring the length of a table with a meter stick, for example. Others — for example, scientific measurements — can require elaborate equipment and complicated techniques.
Is it possible to measure objects without using standard units?
Yes. Nonstandard units (i.e., units that are not agreed upon by large numbers of people) can be used to make comparisons and order objects. But because the units are nonstandard, there is limited value in using them to convey information.
How precise are measurements?
Measurement, by its very nature, is approximate. The precision of the measuring device tells us how finely a particular measurement was made. Measurements made using small units, such as square millimeters, are more precise than measurements made using larger units, such as square centimeters. The accuracy of a measure is determined by how correctly a measurement has been made. Accuracy can be affected by the person making the measurement and/or by the measurement tool. Precision and accuracy, and how to determine them, will be covered in later sessions.
Okay, then — how large is my rock?
It all depends on how you define the word large. Your answer will be based on the attributes you decide to consider, such as weight, volume, surface area, and height.
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.