Learning Math: Measurement
What Does It Mean To Measure? Part A: Comparing Rocks (15 minutes)
Session 1, Part A
Measurement is used in all aspects of daily life, as well as in such fields as engineering, architecture, and medicine. We measure things every day. This morning you may have weighed yourself, poured two cups of water into the coffeemaker, checked the temperature outside to help you decide what to wear, cut enough gift wrap off the roll to wrap a present, decided on the size of a storage container for some leftover food, noted on your car’s odometer how far you’d driven, monitored both your car’s traveling speed and its gas gauge, and kept an eye on the time so that you wouldn’t be late.
All of the situations above are easily identifiable as measurement situations. Yet what is at the heart of all of these comparisons? In other words, in order to measure, what must we consider, and then what steps must we take?
To begin thinking about measurement, you will use, of all things, a rock.
Make a list of attributes that could be used to describe the rock.
Some of these attributes might be measurable, and some might not. How do we determine what we can measure?
If you are having difficulty sorting the attributes, consider which attributes can be quantified. For example, the texture of a rock (e.g., smooth, bumpy, rough) isn’t quantifiable using any of the standard units we know; in contrast, the weight of the rock is quantifiable and can be measured in ounces or grams.
Another suggestion is to see what happens when you combine your object with another, similar object. If the attribute is measurable, then it will increase when the objects are combined. For example, when you combine two rocks, the texture won’t increase or change in any way, but the weight certainly will.
If you were to compare different rocks using each of the measurable attributes you listed in Problem A1, what units would you use?
How could you measure these properties?
Think about what instruments, devices, or methods you might use.
Part A adapted from Chapin, Suzanne, and Johnson, A. Math Matters: Understanding the Math You Teach, Grades K-6. p. 177. © 2000 by Math Solutions, Publications. Used with permission. All rights reserved.
Though we all use measurement daily, most adults have not considered the properties of an object that make it measurable in some way. This first activity is designed to focus attention on the many attributes that are used to compare objects (in this case, rocks) and to sort the attributes into two categories — those that are measurable and those that are not.
Answers will vary. Some answers might be the rock’s length, surface area, volume, weight, color, and texture.
A measurable property is a property that can be quantified using some kind of unit as a basis. For example, length is measurable, since there is a unit of length (an inch, a centimeter, etc.) and we are counting or measuring the number of units in our object. A non-measurable property is one without a standard unit. When we combine objects with a measurable property, the property must increase.
If we wanted to measure some of the properties not commonly measured, we would have to invent a method to do it. For example, to measure texture, we could look at the curvature over the small areas of the object; if the curvature doesn’t change much, we could say that the texture of the object was smooth. Some interesting modern research in mathematics focuses on such “nonstandard” measurements.
Answers will vary. For those listed in our solution to Problem A1, we can measure length in centimeters, surface area in square centimeters, volume in cubic centimeters, and weight in grams.
Answers will vary. We could measure the length of the rock using a ruler or tape measure. We could measure the weight using a scale, the volume using a beaker of water (for displacement), and the surface area using tinfoil.
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.