Students at these grade levels continue to profit from activities that involve the act of measurement. Some of these activities will require students to use two or more units to measure an object. Other activities may require them to use a "broken" ruler whose end is missing. Such activity-oriented explorations nurture students' insights into the process of measurement and into the usefulness and power of mathematics in solving problems. In addition, such activities strengthen students' estimation and higher-order thinking skills.

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. Students 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 operations with numbers that yield 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 expand their understanding of measurement to include new types of measures, especially those involving indirect measurement. For example, they learn about density and force in science class and how these characteristics are measured. Middle school students also should develop a deeper understanding of the concept of rate, experiencing and seeing 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 to recognize and appreciate the use of measurement concepts in other real-world settings.

Building upon K-6 expectations, experiences in grades 7-8 will be such that all students:

H. estimate, make, and use 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 go between two bricks, make a scale drawing of it, and place weights on the bridge until it breaks, noting how much weight it held.

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

• 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 use a scale drawing of an apartment (1/4 inch = 1 foot) to find out how many square yards of carpeting are needed for the rectilinear living room.

• Students measure a given line and compare their results, focussing on the idea that any measurement is approximate. They discuss how accurate their measurements are (degree of precision) and compute the measurement error.
• Each student in a group measures the circumferences and diameters of several round objects. They compare their measurements and decide what the most accurate measurement is. They then find the ratio of the circumference to the diameter for each object.

• Students develop the 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 with the same base length and height on their geoboards. They sketch each parallelogram and record its area (found by counting squares). They then discuss their results.

• 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 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 a building.

• Students measure their hand span in centimeters and then measure the width of their desk in hand spans. They use this information to find the width of their desk in centimeters.
• 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 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:
  
  1000 ml weighs 1 kg
  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 C is about 70 degrees F (room temperature)

• Students use strips of paper to make columns with differently shaped cross-sections. They find the cross-sectional area of each column and then test the columns to see which hold the most weight.
• Students investigate how much water clings to them when they step out of a bath or shower by finding the surface area of their body in square centimeters and multiplying by 0.05 to find the volume in cubic centimeters, since a film of water about 0.05 cm thick clings to the skin.. In order to find their surface area, they use geometric objects such as cylinders and spheres to approximate a person.
• Students investigate the density of oranges by weighing an orange, submersing it in water and measuring the volume of water displaced to find the volume of the orange, and dividing the weight by the volume to find the density. They repeat their experiment with peeled oranges.

• 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.