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.

In grades seven and eight, students begin to look at the measurement process more abstractly while continuing to develop their actual measurement skills and using measurement in connection with other subjects and other topics in mathematics.

All measurement activities should involve both estimation and actual measurement at these grade levels. Estimation strategies should include (1) having a model or referent (e.g., a doorknob is about one meter from the floor), (2) breaking an object to be estimated into parts that are easier to measure (chunking), and (3) dividing the object up into a number of equal parts (unitizing). Students should also discuss when an estimate is appropriate and when an actual measurement is needed and they should have opportunities to select appropriate measuring tools and units.

Especially in the context of making measurements in connection with other disciplines, the approximate nature of measure is an aspect of number that needs particular attention. Because of students' prior experience with counting and with using numerical operations to obtain exact answers, it is often difficult for them to develop the concept of the approximate nature of measuring. Only after considerable experience do they recognize that when they correctly measure to the nearest "unit," the maximum possible error would be one-half of that unit. Teachers must help students to understand that the error of a measurement is not a mistake but rather a result of the limitations of the measuring device being used. Only through measurement activities can students discover and discuss how certain acts, such as the selection and use of measuring tools, can affect the degree of precision and accuracy of their measurements.

Students in grades seven and eight should expand their understanding of measurement to include new types of measures, especially those involving indirect measurement. For example, they learn about density and force and how these characteristics are measured in science class. Middle school students also should develop a deeper understanding of the concept of rate, by experiencing and discussing different rates. Constructing scale drawings and scale models or relating biological growth and form provide excellent opportunities for students to use proportions to solve problems, as does using a variety of measuring tools to find the measures of inaccessible objects. Such personal experiences help students recognize and appreciate the use of measurement concepts in other real-world settings.

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 7 and 8.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 7-8 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 trapezoid. Then they use a transparent grid and count squares to find the area. They compare that result to the area of a rectangle whose base is the average of the two bases of the trapezoid and whose height is the same as that of the trapezoid. They look for a pattern in their results and compare their results to their estimates.

• Students build a bridge out of paper to connect two bricks and place weights on the bridge until it breaks, noting how much weight it held.

• Students read and discuss sections of This Book Is About Time by Marilyn Burns. Its many engaging activities and experiments are interspersed with an historical treatment of time and the instruments designed to measure it.

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

• Students use objects shown in a movie poster for King Kong to determine how tall the ape is.

• As a long-term assessment project, students make a three-dimensional scale model of their classroom.

• Students use a map to plan an auto trip across the United States, finding the distance traveled each day and the amount of time required to drive each day's route.

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

• Students plan a school 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 make and use a scale drawing (1/4 inch = 1 foot) of an apartment and use scale models of the furniture to furnish the living room and dining room.

• Students make a floor plan for a small restaurant furnished with round tables.

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

• Students measure a given hallway in school and compare their results, noting that their results are different because any measurement is approximate. They discuss how accuratetheir individual measurements are (degree of precision) and, after reviewing all of their measurements, determine the likely errors in their individual measurements. They also discuss how more precise measures may be obtained and what degree of precision is needed in this situation.

• Each student in a group measures the circumferences and diameters of several round objects using a tape measure or ruler and string. They compare their measurements and decide what the most accurate set of measurements is for each of the objects. They use a calculator to find the ratio of the circumference to the diameter for each object.

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

• Students develop a formula for finding the surface area of a rectangular prism by constructing boxes of various sizes using graph paper, finding the area of each side and adding them, and looking for patterns in their results. They describe their findings in their journals.

• Students construct different parallelograms whose base and height have the same length on their geoboards. They sketch each parallelogram and record its area (found by counting squares). They discuss their results.

• Students work through the Sketching Similarities lesson that is described in the First Four Standards of this Framework. They use a computer program and measure the sizes of the corresponding sides and corresponding angles of similar figures. They conclude that similar figures have equal corresponding angles and their corresponding sides have the same ratio.

• Students use plastic models of a pyramid and a prism, each having the same height and polygonal base, to investigate the relationship between their volumes.

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

• Students and parents in the McKinley School have been dropping pennies into the very large plastic cylinder in the school lobby in an effort to raise money for new playground equipment. The students are challenged to devise a method to estimate how much money is in the cylinder as a function of the height of the pennies at any given time.

• Students construct a measuring tool that they can use to find the height of trees, flagpoles, and buildings, using cardboard, graph paper, straws, string, and washers.

• Students use proportions to find the height of the flagpole in front of the school.

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

• Students are given a ring and asked to find the height of the person who lost the ring. They measure their own fingers and their heights, plotting the data on a coordinate graph. They use a piece of spaghetti to fit a straight line to the plotted points and make a prediction about the height of the person who owns the ring, based on the data they have collected.

• Students plan the weird name Olympics by renaming standard events in different measurement units. For example, "the hundred meter dash" becomes the "100,000 millimeter marathon" and the "ten meter dive" could become the "onehundredth of a kilometer splash."

• Students continue to use approximate "rules of thumb" learned in earlier grades 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

  1 inch is about 2.5 centimeters

  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 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, and a year.

• Students are told that when people get out of a bath, a film of water about 0.05 cm thick clings to their skin. They are then challenged to find what volume of water clings to the skin of an average eighth grader. In order to make the estimate, of course, they need to estimate the surface area of the body. They can do so by considering a collection of cylinders and spheres that approximates a human.

• Students investigate the concept of density by finding objects for which they can find both the volume and the weight, measuring both, and dividing the latter by the former. Interesting objects to use include an orange, a block of wood, a textbook, a rubber sponge ball, and an airfilled rubber ball. Discussions of the results should lead to interesting conjectures about density which can then be confirmed with additional experimentation.

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

• Students use the computer program The Geometric SuperSupposer to explore the relationship in similar triangles between corresponding sides and the perimeters of the triangles. They also analyze the relationship between corresponding sides and the areas of the triangles.

• Students build a "staircase" using wooden cubes. Then they double all of the dimensions and compare the number of cubes used in the second staircase to the number used in the original staircase.

• Students work through the Rod Dogs lesson that is described in the First Four Standards of this Framework. They discover that if an object is enlarged by a scale factor, its new surface area is the scale factor squared times the original area, and its new volume is the scale factor cubed times the original volume.

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

• Students use straw and string to construct models of the five regular polyhedra: the cube, the tetrahedron, the octahedron, the icosahedron, and the dodecahedron.

• Students use cardboard and tape to construct a model that demonstrates that the volume of a pyramid is one-third that of a prism with the same base and height.

• Students build scale models of the classroom, the school, or a monument.

References

Burns, Marilyn. This Book is About Time. Boston, MA: Little, Brown & Co., 1987.

Software

The Geometric SuperSupposer. Sunburst Communications.

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.

Owens, D. T., Ed. Research Ideas for the Classroom: Middle Grades Mathematics. New York, NY: MacMillan, 1993.

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.