Measurement is the assignment of a numerical value to an attribute of an object, such as the length of a pencil. At more-sophisticated levels, measurement involves assigning a number to a characteristic of a situation, as is done by the consumer price index. Understanding what a measurable attribute is and becoming familiar with the units and processes that are used in measuring attributes is a major emphasis in this Standard. Through their school experience, primarily in prekindergarten through grade 8, students should become proficient in using measurement tools, techniques, and formulas in a range of situations.

The study of measurement is important in the mathematics curriculum from prekindergarten through high school because of the practicality and pervasiveness of measurement in so many aspects of everyday life. The study of measurement also offers an opportunity for learning and applying other mathematics, including number operations, geometric ideas, statistical concepts, and notions of function. It highlights connections within mathematics and between mathematics and areas outside of mathematics, such as social studies, science, art, and physical education.

Measurement lends itself especially well to the use of concrete materials. In fact, it is unlikely that children can gain a deep understanding of measurement without handling materials, making comparisons physically, and measuring with tools. Measurement concepts should grow in sophistication and breadth across the grades, and instructional programs should not repeat the same measurement curriculum year after year. However, it should be emphasized more in the elementary and middle grades than in high school.

A measurable attribute is a characteristic of an object that can be quantified. Line segments have length, plane regions have area, and physical objects have mass. As students progress through the curriculum from preschool through high school, the set of attributes they can measure should expand. Recognizing that objects have attributes that are measurable is the first step in the study of measurement. Children in prekindergarten through grade 2 begin by comparing and ordering objects using language such as longer and shorter. Length should be the focus in this grade band, but weight, time, area, and volume should also be explored. In grades 3–5, students should learn about area more thoroughly, as well as perimeter, volume, temperature, and angle measure. In these grades, they learn that measurements can be computed using formulas and need not always be taken directly with a measuring tool. Middle-grades students build on these earlier measurement experiences by continuing their study of perimeter, area, and volume and by beginning to explore derived measurements, such as speed. They should also become proficient in measuring angles and understanding angle relationships. In high school, students should understand how decisions about unit and scale can affect measurements. Whatever their grade level, students should have many informal experiences in understanding » attributes before using tools to measure them or relying on formulas to compute measurements.

As they progress through school, not only should students' repertoire of measurable attributes expand, but their understanding of the relationships between attributes should also develop. Students in the elementary grades can explore how changing an object's attributes affects certain measurements. For example, cutting apart and rearranging the pieces of a shape may change the perimeter but will not affect the area. In the middle grades this idea can be extended to explorations of how the surface area of a rectangular prism can vary as the volume is held constant. Such observations can offer glimpses of sophisticated mathematical concepts such as invariance under certain transformations.

The types of units that students use for measuring and the ways they use them should expand and shift as students move through the prekindergarten through grade 2 curriculum. In preschool through grade 2, students should begin their study of measurement by using nonstandard units. They should be encouraged to use a wide variety of objects, such as paper clips to measure length, square tiles to measure area, and paper cups to measure volume. Young children should also have opportunities to use standard units like centimeters, pounds, and hours. The "standardization" of units should arise later in the lower grades, as students notice that using Joey's foot to measure the length of the classroom gives a different length from that found by using Aria's foot. Such experiences help students see the convenience and consistency of using standard units. As students progress through middle school and high school, they should learn how to use standard units to measure new abstract attributes, such as volume and density. By secondary school, as students are measuring abstract attributes, they should use more-complex units, such as pounds per square inch and person-days.

Understanding that different units are needed to measure different attributes is sometimes difficult for young children. Learning how to choose an appropriate unit is a major part of understanding measurement. For example, students in prekindergarten through grade 2 should learn that length can be measured using linear tools but area cannot be directly measured this way. Young children should see that to measure area they will need to use a unit of area such as a square region; middle-grades students should learn that square regions do not work for measuring volume and should explore the use of three-dimensional units. Students at all levels should learn to make wise choices of units or scales, depending on the problem situation. Choosing a convenient unit of measurement is also important. For example, although the length of a soccer field can be measured in centimeters, the result may be difficult to interpret and use. Students should have a reasonable understanding of the role of units in measurement by the end of their elementary school years.

The metric system has a simple and consistent internal organization. Each unit is always related to the previous unit by a power of 10: a centimeter is ten times larger than a millimeter, a decimeter is ten times larger than a centimeter, and so forth. Since the customary English system of measurement is still prevalent in the United States, students should learn both customary and metric systems and should know some rough equivalences between the metric and customary systems—for » example, that a two-liter bottle of soda is a little more than half a gallon. The study of these systems begins in elementary school, and students at this level should be able to carry out simple conversions within both systems. Students should develop proficiency in these conversions in the middle grades and should learn some useful benchmarks for converting between the two systems. The study of measurement systems can help students understand aspects of the base-ten system, such as place value. And in making conversions, students apply their knowledge of proportions.

Understanding that all measurements are approximations is a difficult but important concept for students. They should work with this notion in grades 3–5 through activities in which they measure certain objects, compare their measurements with those of the rest of the class, and note that many of the values do not agree. Class discussions of their observations can elicit the ideas of precision and accuracy. Middle-grades students should continue to develop an understanding of measurements as approximations. In high school, students should come to recognize the need to report an appropriate number of significant digits when computing with measurements.

Measurement techniques are strategies used to determine a measurement, such as counting, estimating, and using formulas or tools. Measurement tools are the familiar devices that most people associate with taking measurements; they include rulers, measuring tapes, vessels, scales, clocks, and stopwatches. Formulas are general relationships that produce measurements when values are specified for the variables in the formula.

Students in prekindergarten through grade 2 should learn to use a variety of techniques, including counting and estimating, and such tools as rulers, scales, and analog clocks. Elementary and middle-grades students should continue to use these techniques and develop new ones. In addition, they ought to begin to adapt their current tools and invent new techniques to find more-complicated measurements. For example, they might use transparent grid paper to approximate the area of a leaf. Middle-grades students can use formulas for the areas of triangles and rectangles to find the area of a trapezoid. An important measurement technique in high school is successive approximation, a precursor to calculus concepts.

Students should begin to develop formulas for perimeter and area in the elementary grades. Middle-grades students should formalize these techniques, as well as develop formulas for the volume and surface area of objects like prisms and cylinders. Many elementary and middle-grades children have difficulty with understanding perimeter and area (Kenney and Kouba 1997; Lindquist and Kouba 1989). Often, these children are using formulas such as P = 2l + 2w or A = lw without understanding how these formulas relate to the attribute being measured or the unit of measurement being used. Teachers must help students see the connections between the formula and the actual object. In high school, as students use formulas in solving problems, they should recognize that the units in the measurements behave like variables under » algebraic procedures, and they can use this observation to organize their conversions and computations using unit analysis.

Estimating is another measurement technique that should be developed throughout the school years. Estimation activities in prekindergarten through grade 2 should focus on helping children better understand the process of measuring and the role of the size of the unit. Elementary school and middle-grades students should have many opportunities to estimate measures by comparing them against some benchmark. For example, a student might estimate the teacher's height by noting that the teacher is about one and one-half times as tall as the student. Middle-grades students should also use benchmarks to estimate angle measures and should estimate derived measurements such as speed.

Finally, students in grades 3–5 should have opportunities to use maps and make simple scale drawings. Grades 6–8 students should extend their understanding of scaling to solve problems involving scale factors. These problems can help students make sense of proportional relationships and develop an understanding of similarity. High school students should study more-sophisticated aspects of scaling, including the effects of scale changes on a problem situation. They should also come to understand nonlinear scale changes such as logarithmic scaling and how such techniques are used in analyzing data and in modeling.