Students can develop a strong understanding of measurement and measurement systems from consistent experiences in classroom activities where a variety of manipulatives and technology are used. The key components of this understanding, as identified in the K12 Overview, are: the concept of a measurement unit; standard measurement units; connections to other mathematical areas and to other disciplines; indirect measurement; and, for older students, measurement error and degree of precision.

Building upon the measurement skills and understandings developed in grades K-8, high school students move to a more routine use of measurement. They examine measurement as a more abstract process, focusing on measurement error and degree of precision. They spend much more time on indirect measurement techniques than they did in earlier grades, expanding their repertoire to include not only the use of proportions and similarity but also the use of the Pythagorean Theorem and basic right triangle trigonometric relationships.

Students at the high school level will frequently use measurement to help develop connections to other mathematical concepts. For example, students may use a computer program that measures angles to help them discover the relationship between the measures of two vertical angles formed by intersecting lines or the measures of inscribed angles intercepting the same arc of a circle. They may also develop algebraic techniques to help them find measures, as, for example, when they develop a formula for finding the distance between two points in the coordinate plane.

High school students also use measurement frequently in connection with other subject areas. Science experiments require a precise use of measurement. Social studies activities often require students to read and interpret maps and/or scale drawings. In technology classes, woodshop, drafting, sewing, and cooking, students must also use a variety of measuring tools and techniques; and in physical education, students frequently will need to measure distances and rates.

The cumulative progress indicators for grade 12 appear below in boldface type. Each indicator is followed by activities which illustrate how it can be addressed in the classroom in grades 9, 10, 11, and 12.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 9-12 will be such that all students:

17. Use techniques of algebra, geometry, and trigonometry to measure quantities indirectly.

• Students use coordinate geometry techniques to determine the distance between two points.

• Students use similar figures and proportions to measure the height of a tree or a flagpole.

• Students use the Pythagorean Theorem to determine how long a ladder is needed to climb a wall, including a determination of a safe angle at which to place the ladder.

• Students use right-triangle trigonometry to measure the width of a canyon or the height of a waterfall.

• Students work through the Ice Cones lesson that is described in the First Four Standards of this Framework. They use the formula for the volume of a cone and a graphing calculator to determine the maximum volume of a cone made from a paper circle of radius 10 cm which is cut along a radius.

18. Use measurement appropriately in other subject areas and career-based contexts.

• Students investigate how the volume of a cereal box changes with its area by finding the volume and surface area of a box of their favorite kind of cereal. They also discuss how the shape of the box affects its volume and surface area and why the volume of the box is so large for the amount of cereal it contains.

• As a question on their takehome final exam in Algebra, students are asked to measure their own and gather data about the length of other people's femurs and overall height in an effort to determine whether there is a relationship between the two lengths. They plot the resulting ordered pairs on a coordinate plane and find a line of best fit for their data. For extra credit, they make a prediction of the height of a male with a 22 inch femur.

• Students discuss the possible meaning of "light year" as a unit of measure and find units equivalent to it.

19. Choose appropriate techniques and tools to measure quantities in order to achieve specified degrees of precision, accuracy, and error (or tolerance) of measurements.

• Students use significant digits appropriately in measuring large distances, such as the distance from one school to another, from one city to another, and from one planet to another.

• Students find the distance between two cities by adding the numbers given on a road map for the segments of the trip, by measuring the segments and using the mileage scale, and by referring to a published mileage table. They explain the different results by referring to the degrees of prevision of the different measurements.

• In making a scale drawing of a house, students discuss the degree of accuracy of their measurements.

• Students read and discuss the photographs in Powers of Ten by Phillip and Phylis Morrison and the office of Charles and Ray Eames, and view the associated videotape. This wellknown book takes the reader on a trip through perspectives representing fortytwo powers of ten, from the broadest view of the universe to the closest view of the nucleus of an atom. The measurement units used and the progression from one to another highlight the range and power of our system of measurement.

• Students use computer drawing and measuring utilities to discover geometric concepts. They also discuss the limitations of such a program. For example, a program may give 14.7 for the length of the base of a triangle and 7.3 for its midline (the segment joining the midpoints of the other two sides); however, because of the program's measurement limitations, its answer for the length of the midline may not be exactly half the length of the base, as is the case in reality.

• Students determine what kind of measuring instrument needs to be used to measure ingredients for pain-relievers, for cough syrup, for a cake, and for a stew. They bring to class a variety of empty bottles and packages and note how the ingredients are measured: What does 325 mg (of acetaminophen) mean, or one fluid ounce (of cough syrup) as opposed to 1/4 cup of oil (for a cake). They discuss the accuracy, error, and tolerance of each measurement.

References

Morrison, Philip, Phylis Morrison, and the office of Charles and Ray Eames. Powers of Ten. Scientific American, 1988. (Revised, 1994. See also reference under Video.)

General reference

Froelich, G. Curriculum and Evaluation Standards for School Mathematics: Addenda Series: Connecting Mathematics. Reston, VA: National Council of Teachers of Mathematics, 1991.

Video

Powers of Ten. Philip Morrison, Phylis Morrison, and the office of Charles and Roy Eames. New York: Scientific American Library, 1991.

One-Line Resources

http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/

The Framework will be available at this site during Spring 1997. In time, we hope to post additional resources relating to this standard, such as grade-specific activities submitted by New Jersey teachers, and to provide a forum to discuss the Mathematics Standards.