In grades 5-8, the mathematics curriculum should include extensive concrete experiences using measurement so that students can--

• extend their understanding of the process of measurement;
• estimate, make, and use measurements to describe and compare phenomena;
• select appropriate units and tools to measure to the degree of accuracy required in a particular situation;
• understand the structure and use of systems of measurement;
• extend their understanding of the concepts of perimeter, area, volume, angle measure, capacity, and weight and mass;
• develop the concepts of rates and other derived and indirect measurements;
• develop formulas and procedures for determining measures to solve problems.

Focus

Measurement activities can and should require a dynamic interaction between students and their environment. Students encounter measurement ideas both in and out of school, in such areas as architecture, art, science, commercial design, sports, cooking, shopping, and map reading. The study of measurement shows the usefulness and practical applications of mathematics, and students' need to communicate about various measurements highlights the importance of standard units and common measurement systems.

Measurement in grades 5-8 should be an active exploration of the real world. As students acquire the ability to use appropriate tools in measuring objects, they should extend these skills to new situations and new applications. The approximate nature of measure is an aspect of number that deserves repeated attention at this level. However, measurement activities in these grades should focus on using concepts and skills to solve problems and investigate other mathematical situations.

The development of the concepts of perimeter, area, volume, angle measure, capacity, and weight is initiated in grades K-4 and extended and applied in grades 5-8. At this level, students can begin to estimate the error of a measurement, adding to the K-4 notion of "about" 4 cm. From their explorations, students should develop multiplicative procedures and formulas for determining measures. The curriculum should focus on the development of understanding, not on the rote memorization of formulas. In addition, the concepts of rate as a measure and of indirect measurement are developed in grades 5-8.

Geometry and measurement are interconnected and support each other in many ways. The concept of similarity, for example, can be used in indirect measurement, and the perimeter and area of irregular figures can be determined using line segments and squares, respectively. Measurement also has strong connections to the students' expanding concept of number. Fractions, decimals, and rational numbers are used to represent measures.

Discussion

In everyday life, people need to make many kinds of measures to resolve common questions: About how long will it take? About how much do I need to buy? About how much will it hold? An estimate is often sufficient. Estimation requires a judgment about an entity's approximate relationship to a standard. Students' skills at estimating measurements will develop only through experience. One important aspect of estimating measurements is context. Students need to develop estimation strategies, and they need experience in judging what degree of accuracy is required in a given situation. If a person is buying carpet, error should be in the direction of an overestimate. However, if one is estimating how much time to sunbathe without burning, an underestimate is best. In developing estimation skills for measurement, a student learns to relate the world to familiar personal experiences. The ability to hold one's hands about a meter apart, to know the length of a foot or stride, to know the width of a fingernail--all these are useful estimating tools.

During students' early experiences with counting and operations using whole numbers, they work with precise situations that yield exact counts. Measuring the length of an object is quite different, and it is essential that students understand this difference. The approximate nature of measuring is a concept that takes time and many experiences for students to develop and understand. The following classroom activity helps students with this concept and relates to the standard on statistics.

Have each student use a meter tape to measure the length of the room to the nearest centimeter. Record each student's measure and analyze the results (see fig. 13.1).

Fig. 13.1. Data plots

Linked to the development of measuring concepts are experiences with standard measuring tools: rulers, balances, protractors, clocks, wheels, speedometers, and so on. In a given situation, a student must select both an appropriate unit and a tool to find a measurement; this selection depends on the degree of accuracy required in a particular situation. It would be inappropriate to select a 10-cm ruler to measure the length of a soccer field, even when a fairly accurate measure is needed. However, the square corner of a sheet of paper can be used to "measure" an angle if one only needs to know whether it is larger or smaller than a right angle.

As students progress through grades 5-8, they should develop more efficient procedures and, ultimately, formulas for finding measures. Length, area, and volume of one-, two-, and three-dimensional figures are especially important over these grade levels. For example, once students have discovered that it is possible to find the area of a rectangle by covering a figure with squares and then counting, they are ready to explore the relationship between areas of rectangles and areas of other geometric figures. This exploration gives students an opportunity to reason deductively and see how mathematical ideas relate to one another. The following sketches suggest some possibilities.

The area of a parallelogram can be rearranged into a rectangle (fig. 13.2).

Fig. 13.2. Parallelogram to rectangle

The area of a triangle is one-half the area of a parallelogram (fig. 13.3).

Fig. 13.3. Triangle to parallelogram

All polygons can be partitioned into triangles (fig. 13.4).

Fig. 13.4. Partitioning polygons into triangles

All these connections require students to understand that the area of a figure does not change if it is partitioned and rearranged. It is also important that students understand the association between multiplication and determining the area of a rectangle. The formula is not a "magic box." It is a summary of a process that tells how many units it takes to cover the rectangle. It is also a summary of the relationship among area, height, and length. Any two of these determine the other: A = LH; L = A/H; H = A/L.

An example of a practical problem that involves measurement, similarity in scale drawing, and creativity is the following (Wirszup and Streit 1987).

Given a piece of plywood 150 cm x 300 cm, design a dog kennel that can be made from the piece. Try to make your kennel as large as you can. Make a scale drawing to show how the parts of the kennel have to be cut from the plywood. Give the measurements. Draw a sketch or sketches to show what the finished kennel will look like. Write the measurements on the sketches.

Students need many experiences with the concepts of rate in measurement settings. Here is an example of a problem that uses rates as measures (Meyer and Sallee 1983):

It is the seventh annual cross-country motorcycle race across the Nevada desert, 70 miles and back. Orite, on her new Harley-Davidson, averages 80 miles an hour going out but has clutch trouble and can manage only 60 miles an hour coming back. Eric, on a Honda, can go only 70 miles an hour, but he keeps it up for the entire race. Who wins the race?

Constructing a scale model at the solar system is another problem that involves proportional reasoning and connects mathematics to another discipline. The gymnasium or the hall of the school can be used. Students have to decide what will represent the orbit of Pluto and then figure out what the radius of the other orbits will be. See figure 13.5.

Fig. 13.5. Solar system

Areas of irregular figures can be approximated by covering the figure with a square grid and counting the whole squares within the figure as an inner measure and all squares that touch the figure anywhere as an outer measure. The actual measure is between these two, so the mean of the measures gives an estimate of the area and half the difference between the measures gives the greatest possible error. If the possible error is too big, the process can be repeated with a smaller grid. See figure 13.6.

Fig. 13.6. Estimating area

Students can use their knowledge of similar triangles to measure heights of inaccessible objects. Two possibilities are illustrated in figure 13.7, one using shadows and the other using reflections in a mirror.

Fig. 13.7. Indirect measurement