Learning Math: Measurement
The Metric System Homework
Session 3, Homework
- The meter was originally based on the size of the Earth, with the distance from the equator to the North Pole being arbitrarily defined as 10 million m. What is another way to express the distance of 10 million m?
- The Earth is not quite spherical, but for practical purposes we can think of Earth as having a circumference of 40 Mm. Thus, originally the meter was considered to be about 1/40,000,000 of Earth’s circumference. Use the Web or reference books to find out how a meter is officially defined today.
Match the metric quantities on the left with the approximate lengths/distances on the right:
|1 gigameter||(1 • 109)||A. distance a fast walker walks in 10 minutes|
|1 megameter||(1 •106)||B. size of an atom|
|1 kilometer||(1 •103)||C. waist height of an average adult|
|1 meter||(base unit)||D. size of bacteria|
|1 centimeter||(1 •10-2)||E. thickness of a dime|
|1 millimeter||(1 •10-3)||F. distance from Atlanta to Miami|
|1 micrometer||(1 •10-6)||G. width of a fingernail|
|1 picometer||(1 •10-12)||H. Earth’s distance from Saturn|
A nickel is said to weigh 5 g. How much is 1 kg of nickels worth?
Give the approximate mass of the following volumes of water:
- 6.5 L
- 30 cm3
- 18 mL
- 12 m3
Why might a student be confused by this question: Which is more, 1.87 kg or 1,869 g? Explain.
The article “Do Your Students Measure Up Metrically?” points out some of the challenges of helping students in the United States learn the metric system. Discuss or think about how you might improve instruction on the metric system in your classroom or school.
Taylor, P. Mark; Simms, Ken; Kim, Ok-Kyeong; and Reys, Robert E. (January, 2001). Do Your Students Measure Up Metrically? Teaching Children Mathematics, 282-287.
Reproduced with permission from Teaching Children Mathematics. © 2001 by the National Council of Teachers of Mathematics. All rights reserved.
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Do Your Students Measure Up Metrically?
- Ten million meters can also be expressed as 10,000 km, or 10 Mm.
- A meter is defined to be the length of the path traveled by light in vacuum during a time interval of 1/299,792,458 of a second. In this way, it can be defined in terms of the second, another base unit of the metric system, and the constant speed of light.
- 1 gigameter – H. Earth’s distance from Saturn
- 1 megameter – F. distance from Atlanta to Miami
- 1 kilometer – A. distance a fast walker walks in 10 minutes
- 1 meter – C. waist height of an average adult
- 1 centimeter – G. width of a fingernail
- 1 millimeter – E. thickness of a dime
- 1 micrometer – D. size of bacteria
- 1 picometer – B. size of an atom
Since it takes 1,000 g to make a kilogram, there are about 200 nickels in a kilogram. Two hundred nickels are worth $10.
- The mass is approximately 6.5 kg.
- Since 1 cm3 is equivalent to 1 g, 30 cm3 of water has a mass of 30 g.
- Since 1 mL is equivalent to 1 g, 18 mL has a mass of 18 g.
- Twelve cubic meters is equivalent to 12,000 dm3. Since 1 dm3 of water has a mass of 1 kg, 12 m3 has a mass of 12,000 kg.
In terms of the number of units, 1,869 is a much larger number, so a student might be confused and say that 1,869 g is more. But since a kilogram is equivalent to 1,000 g, 1.87 kg is actually 1,870 g, which is more. This confusion might disappear once the student is more familiar with the metric system, which makes this type of conversion much easier than converting, say, inches to miles.
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.