Learning Math: Measurement
The Metric System Part A: Metric System Basics (35 minutes)
Session 3, Part A
In This Part
- Units and Prefixes
- Working With Metric Prefixes
The metric system was introduced in France in the 1790s as a single, universally accepted system of measurement. But it wasn’t until 1875 that the multinational Treaty of the Meter was signed (by the United States and other countries), creating two groups: the International Bureau of Weights and Measures and the international General Conference on Weights and Measures. The purpose of these groups was to supervise the use of the metric system in accordance with the latest pertinent scientific developments.
Since then, the Bureau and Conference have recommended improvements in terms of the accuracy and reproducibility of metric units. In fact, by 1960, improvements were so great that a “new” metric system, called the International System of Units (abbreviated SI), was created. The metric system we use today is actually SI. Although there are some discrepancies between the two systems, these differences are slight from a nonscientific perspective; therefore, we will refer to SI as “the metric system.”
Let’s begin by reviewing the metric system. Note 3
Make a list of some of the facts, relationships, and symbols you recall about the metric system.
Why do you think most countries use the metric system?
Units and Prefixes
One of the strengths of the metric system is that it has only one unit for each type of measurement. Other units are defined as simple products or quotients of these base units.
For example, the base unit for length (or distance) is the meter (m). Other units for length are described in terms of their relationship to a meter: A kilometer (km) is 1,000 m; a centimeter (cm) is 0.01 of a meter; and a millimeter (mm) is 0.001 of a meter.
Prefixes in the metric system are short names or letter symbols for numbers that are attached to the front of the base unit as a multiplying factor. A unit with a prefix attached is called a multiple of the unit — it is not a separate unit. For example, just as you would not consider 1,000 in. a different unit from inches, a kilometer, which means 1,000 m, is not a different unit from meters.
Use the chart below to explore some common units and prefixes in the metric system. Pay attention to the patterns that emerge as you look at the different prefixes.
Working With Metric Prefixes
Here is a detailed table of some metric prefixes: Note 4
This session focuses only on the metric system, but you may also want to use this session to review what you know about the customary system. The concept of a coherent system may be confusing. If the customary system were coherent (which it is not), then there would be a single base unit for length; area and volume would also be based on this unit. For example, if the inch were the base unit, then we would measure area in square inches (not in acres or square yards) and volume in cubic inches (not in pints or cubic feet). In a coherent system, units can be manipulated with simple algebra rather than by remembering complex conversion factors.
If you are working in a group that isn’t very familiar with the metric system, you can take this opportunity to practice expressing quantities in more than one way. For example, working in pairs or as a class, participants can make up questions for one another to answer and/or quantities to express using different prefixes. Refer to Problems A3-A5 for guidance.
Answers will vary. Some of the common measurements are the meter, the liter, and the gram. Most people associate the number 10 with the metric system, since relationships in this system are determined by powers of 10.
Most countries use the metric system because it is an international standard; it makes it easy for different countries to talk about the same units of length or mass. The metric system is also designed for easy calculation.
- There are many patterns in the table that you may want to discuss. First, the metric system uses prefixes in front of a unit name as a multiplying factor. Most prefixes multiply or divide units in steps of 1,000 as you go up and down the table.
- The more commonly used prefixes are those from 10-3 to 103. These units are commonly used when measuring areas, volumes, and lengths. Throughout this session, you should focus on gaining familiarity with those particular prefixes.
- A centimeter is a 100th of a meter (1 m = 100 cm). A millimeter is a 1,000th of a meter (1 m = 1,000 mm). A kilometer is 1,000 m (1,000 m = 1 km). A micrometer is a millionth of a meter (1 m = 1,000 000 μm).
- These measurements are related because they are all built on powers of 10, with the meter as the base unit. The prefix tells you how many meters (or how much of a meter) you’re talking about.
- A centimeter could be 10 mm or 0.01 m. A millimeter could be 0.1 cm or 0.001 m. A decimeter could be 10 cm, 100 mm, 0.1 m. Other answers are also possible!
One primary reason is that a number written in metric units can easily be converted by multiplying or dividing by powers of 10, since this type of multiplication is done by moving the decimal point. For example, 2.54 cm is equivalent to 25.4 mm and 0.0254 m. Each of these conversions is done by knowing the power of 10 associated with the conversion and moving the decimal point by that many places.
- As 1,000 m is 1 km, 3,600 m is 3.6 km. We simply moved the decimal point three places to the left.
- Since 0.001 m is 1 mm, 0.028 m is 28 mm. Again, we just moved the decimal point three places to the right.
- One way is to convert the millimeters to meters and then to kilometers: 1,000 mm equals 1 m, so 4,600,000 mm equals 4,600 m. Similarly, 1,000 m equals 1 km, so 4,600 m equals 4.6 km (i.e., we moved the decimal point six places to the left).
To make a fair comparison, we must first convert gigameters to megameters (or vice versa). Since 1 Gm = 1,000 Mm, the Sun’s distance from the Earth is 150 • 1,000, or 150,000 Mm. The ratio is now 150,000:384, or about 390 times farther away.
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.