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.

Why teach measurement? The ability to measure enables students to connect mathematics to the environment and offers opportunities for interdisciplinary learning in social studies, geography, science, music, art, and other disciplines. In addition, measurement tools and skills have a variety of uses in everyday adult life.

However, in the most recent international assessment of mathematics achievement, 13-year-olds in the United States performed very poorly in comparison to other nations. The results of this study indicated that, while students are given instruction on measurement, they do not learn the concepts well. For example, some students have difficulty recognizing two fundamental ideas of measurement, the concept of a unit and the iteration of units. A common error is counting number marks on a ruler rather than counting the intervals between the marks. Another difficult concept is that the size of the unit and the number of units needed to measure an object are inversely related; as one increases the other decreases. In the fifth and sixth grades, students begin to encounter both very small and very large standard measurement units (such as milligrams or tons), and these ideas become increasingly critical to understanding measurement.

Students must be involved in the act of measurement; they must have opportunities to use measurement skills to solve real problems if they are to develop understanding. Textbooks by themselves can only provide symbolic activities. Middle grade teachers must take responsibility for furnishing hands-on opportunities that reinforce measurement concepts with all common measures.

Using measurement formulas as a more efficient approach to some types of direct measurement is an important part of fifth and sixth grade mathematics. It represents the first formal introduction to indirect measurement. Multiplying the length by the width of a rectangle is certainly an easier way to find its area than laying out square units to cover its surface. The formulas should develop, however, as a result of the students' exploration and discovery; they should be seen as efficient ways to count iterated units. Having students memorize formulas that, for them, have no relation to reality or past direct measurement experiences will be unsuccessful. Fewer than half of the U.S. seventh grade students tested in the international competition could figure out the area of a rectangle drawn on a sheet of graph paper, and only slightly more than half could compute the area given the dimensions of length and height. Often length and width are taught separately and how the two measurements combine to form the square units of area is not emphasized in instruction. In addition, area and perimeter are often confused with each other by middle grade students. Limiting students' experiences with measurement to the printed pages of textbooks restricts flexibility so that their understanding cannot be developed or generalized.

In order to further strengthen students' understanding of measurement concepts, it is important to provide connections of measurement to other ideas in mathematics and to other areas of learning. Students should measure objects, represent the information gathered visually (e.g., in a graph), model the situation with symbols (e.g., with formulas), and apply what they have learned to real-world events. For example, they might collect information about waste in the school lunchroom and present their results to the principal with suggestions for reducing waste. Integrating across mathematical topics helps to organize instruction and generates useful ideas for teaching the important content of measurement.

In summary, measurement activities should require a dynamic interaction between students and their environment, as students encounter measurement outside of school as well as inside school. Students should use each measuring instrument until its use becomes second nature. The curriculum should focus on the development of understanding of measurement rather than on the rote memorization of formulas. This approach can be reinforced by teaching students to estimate and to be aware of the context whenever they make an estimate. For example, when buying carpeting it is advisable that the estimate be too high rather than too low. Students must be given the opportunity to extend their learning to new situations and new applications.

The cumulative progress indicators for grade 8 appear below in boldface type. Each indicator is followed by activities which illustrate how it can be addressed in the classroom in grades 5 and 6.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 5-6 will be such that all students:

7. Use estimated and actual measurements to describe and compare phenomena.

• Students estimate the number of square centimeters in a triangle. Then they enclose the triangle in a rectangle and use centimeter cubes or a transparent square centimeter grid to find the area of the rectangle. They also count squares to find the area of the triangle and that of any other triangles formed by the rectangle. They look for a pattern in their results and compare their results to their estimates.

• Students explain why the following is or is not reasonable: An average person can run one kilometer in one minute.

• Students measure how long it takes to go 10 meters, first using "baby steps," then using normal steps, and finally using "giant steps." They then compare their rates.

• Students measure the area of their foot by tracing around it on centimeter graph paper and counting the number of squares covered. To ease the counting task, students can color the squares completely inside the outline blue, those that are onehalf inside green, those that are onethird inside yellow, and those that are onefourth inside orange. Then all of the likecolored squares can be counted more easily and the various totals added to each other.

• An interesting openended group assessment project to use after the previous foottracing activity has been completed is to tell the students that Will Perdue (of the Chicago Bulls) wears a size 18 « shoe which measures 21 « inches long. Students are asked to use what they know about the areas of their own feet to estimate the area of his foot. Students who make a good estimate will deal with several issues: the fact that they have information about the length of their own feet, but only about the length of Will's shoe; the fact that as the foot gets longer, it also gets wider; and the issue of how to set up a proportion between appropriate quantities.

8. Read and interpret various scales, including those based on number lines and maps.

• Students use a given scale to compute the actual length of a variety of illustrated dinosaurs.

• Students make a scale drawing of their classroom and use two-dimensional scale models of its furniture in order to propose new ways of arranging the classroom so that they can work more efficiently in cooperative groups.

• Students use a map to find the distance between two cities.

• Students work through the Short-Circuiting Trenton lesson that is described in the Introduction to this Framework. Using a map of Trenton and a ruler, students determine the distances between various sites, and then find the most efficient walking tour for their class trip.

9. Determine the degree of accuracy needed in a given situation and choose units accordingly.

• Students plan a vegetable garden, determining the unit of measure appropriate for the garden, estimating its size, and then computing the perimeter (for fencing) and area (for fertilizer).

• Groups of students use a scale drawing of an apartment (1 cm = 1 foot) to find out how many square yards of carpeting are needed for the rectangular (9' x 12') living room and other rooms.

• Statements work through the Mathematics at Work lesson that is described in the Introduction to this Framework. A parent discusses a problem which her company faces regularly: to determine how large an air conditioner will be needed for a particular room. To solve this problem, the company has to estimate the size of the room, in terms of its volume and the areas of any windows, after determining the appropriate units.

10. Understand that all measurements of continuous quantities are approximate.

• Students measure a specific distance in the room and compare their results, focusing on the idea that any measurement is approximate.

• Before a cotton ball toss competition, students discuss what units should be used to measure the tosses. They decide that measuring to the nearest centimeter should be close enough, even though the actual tosses will probably be slightly more or slightly less.

11. Develop formulas and procedures for solving problems related to measurement.

• Students complete a worksheet showing several rectangles on grids that are partially obscured by inkblots. In order to find the area of each rectangle, they must use a systematic procedure involving multiplying the length of the rectangle by its width.

• Students develop the formula for finding the volume of a rectangular prism by constructing and filling boxes of various sizes with centimeter or inch cubes and looking for patterns in their results. As a journal entry, they describe the "shortcut" way to find the volume.

12. Explore situations involving quantities which cannot be measured directly or conveniently.

• Students work in groups to estimate the number of bricks needed to build the school building. They explain their results in a class presentation, describing the strategies they used.

• Students are asked to estimate how many heads tall they are. Then they work in groups to develop a procedure for finding out how many heads tall each student is.

• Students construct a measuring tool that they can use to find the height of trees, flagpoles, and buildings when they are standing a fixed distance from the object to be measured.

13. Convert measurement units from one form to another, and carry out calculations that involve various units of measurement.

• Students are asked to find how many pumpkin seeds there are in a kilogram. They decide to measure how much 50 seeds weigh and use this result to help them find the answer.

• Students use approximate "rules of thumb" to help them convert units. For example:

  1 km is about 6/10 of a mile

  1 liter is a little bigger than a quart

  1 meter is a little bigger than a yard

  1 kg is about 2 pounds

  20 degrees Centigrade is about 70 degrees Fahrenheit (room temperature)

  1000 ml of water normally weighs about 1 kg

14. Understand and apply measurement in their own lives and in other subject areas.

• Students measure the heights of bean plants at regular intervals under different conditions. Some are in sunlight and some are not. The students discuss their results and make a graph of their findings.

• Students estimate and weigh cups filled with jellybeans, raisins, dried beans, peanuts, and sand to find out that equal volumes of different objects do not always weigh the same.

• Students learn how much water is in different foods by first trimming pieces of 5 different foods to a standard 15 grams, then measuring their weights again the next day. Where did the water go?

• Students estimate what fraction of an orange is edible, then weigh oranges, peel them and separate the edible parts. They weigh the edible part and then compute what fraction is actually edible and compare that fraction to their estimate.

• Students create their own food recipes.

• An ice cube is placed on a plastic tray in five different parts of the classroom. One group of students is assigned to each ice cube and tray. The students are asked to estimate how long it will take each ice cube to melt. They then observe the ice cube at five-minute intervals, recording their observations. After the ice cubes have melted, the groups share their observations and compare the length of time it took for the ice cubes to melt. They make a conjecture about the warmest spots in the classroom and then measure the temperature in each location to confirm their conjecture.

• Students read Anno's Sundial by Mitsumasa Anno. This sophisticated threedimensional popup book presents an extraordinary amount of information about the movement of the earth and the sun, the relationship between those movements, and how people began to tell time. It is an ideal kickoff for an integrated, multidisciplinary unit with upperelementary students, incorporating reading, social studies, science, and mathematics.

• Students estimate and then develop a plan to find out how many pieces of popped popcorn will fit in their locker.

• Students work in pairs to design a birdhouse that can be made from a single sheet of wood (posterboard) that is 22" x 28". The students use butcher paper to lay out their plans so that the birdhouse is as large as possible. Each pair of students must show how the pieces can be laid out on the posterboard before cutting.

• Students compare the measurements of an object to those of its shadow on a wall as the distance between the object and the wall increases.

15. Understand and explain the impact of the change of an object's linear dimensions on its perimeter, area, or volume.

• Students use pattern blocks to see how the area of a square changes when the length of its side is doubled. They repeat the experiment using equilateral triangles.

• Students use cubes to explore how the volume of a cube changes when the length of one side is doubled, then when the lengths of two sides are doubled, and, finally, when the lengths of all three sides are doubled.

• Students use graph paper to draw as many rectangles as they can that have a perimeter of 16 units. They find the area of each rectangle, look for patterns, and summarize their results.

16. Apply their knowledge of measurement to the construction of a variety of two- and three-dimensional figures.

• Students use paper fasteners and tagboard strips with a hole punched in each to investigate the rigidity of various polygon shapes. For shapes that are not rigid, they determine how they can be made so.

• Students design and carry out an experiment to see how much water is wasted by a leaky faucet in an hour, a day, a week, a month, a year.

• Students use straw and string to construct models of the two simplest regular polyhedra, the cube and the tetrahedron.

References

Anno, Mitsumasa. Anno's Sundial. New York: Philomel Books, 1987.

General references

Geddes, D. Curriculum and Evaluation Standards for School Mathematics: Addenda Series: Measurement in the Middle Grades. Reston, VA: National Council of Teachers of Mathematics, 1991.

Bright, G. W., and K. Hoeffner. "Measurement, Probability, Statistics, and Graphing," in D. T. Owens, Ed. Research Ideas for the Classroom: Middle Grades Mathematics. New York: Macmillan Publishing Company, 1993.

LaPoint, A. E., N. A. Mead, and G. W. Phillips. A World of Differences: An International Assessment of Mathematics and Science. Princeton, NJ: Educational Testing Service, 1989.

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.