Meaning and Importance

This standard is also, in many ways, the prototypical "integrated" standard because of its strong and essential ties to almost every one of the other content standards. Measurement is an ideal context for dealing with numbers and numerical operations of all sorts and at all levels. Fractions and decimals especially appear very naturally in real-world measurement settings. In fact, metric measures provide perhaps the most useful real-world model of a base-ten numeration system we can offer to children. Geometry and measurement are similarly almost impossible to think about separately. Very similar treatments of area and perimeter, for instance, are called "measurement" topics in some curricula and "geometry" topics in others because they are, quite simply, measurements of geometric figures. Yet another of the content standards which is inextricably linked to measurement is estimation. Estimation of measures should be one of the focuses of any work that children do with measurement. Indeed, the very concept that any measurement is inexact -- is at best an "estimate" -- is a concept that must be developed throughout the grades.

Think about how many different content standards are incorporated into one simple measurement experience for middle grades students: the measurement of a variety of circular objects in an attempt to explore the relationship between the diameter and circumference of a circle. Clearly involved are the measurement and geometry of the situation itself, but also evident are opportunities to deal with patterns in the search for regularity of the relationship, estimation in the context of error in the measurements, and number sense and operations in the meaning of the ratio that ultimately presents itself.

K-12 Development and Emphases

Much research has been done into the development of children's understanding of measurement concepts and the general agreement in the findings leads to a coherent sequence in curriculum. Young children start by learning to identify the attributes of objects that are measurable and then progress to direct comparisons of those attributes among a collection of objects. They would suggest, for instance, that this stick is longer than that one or that the apple is heavier than the orange. Once direct comparisons can consistently be made, informal, non-standard units like pennies or "my foot" can be used to quantify how heavy or how long an object is. Following some experiences pointing out the necessity of being able to replicate the measurements, regardless of the measurer or the size of the measurer's foot, these non-standard units quickly give way to standard, well-defined units like grams and inches.

Older students should continue to develop their notions of measurement by delving more deeply into the process itself and by measuring more complex things. Dealing with various measurement instruments, they should be asked to confront questions concerning the inexact nature of their measures, and to adjust for, or account for, the inherent measurement error in their answers. Issues of the degree of precision should become more important in their activities and discussion. They need to appreciate that no matter how accurately they measure, more precision is always possible with smaller units and better instrumentation. Decisions about what level of precision is necessary for a given task should be discussed and made before the task is begun.

Students should also begin to develop procedures and formulas for determining the measures of attributes like area and volume that are not easily directly measured, and also to develop indirect measurement techniques such as the use of similar triangles to determine the height of a flagpole. Their universe of measureable attributes expands to include measures of a whole variety of physical phenomena (sound, light, pressure) and a consideration of rates as measures (pulse, speed, radioactivity).

The growth of technology in the classroom also opens up a wide range of new possibilities for students of all ages. Inexpensive instruments that attach to graphing calculators and computers are capable of making and recording measurements of temperature, distance, sound and light intensity, and many other physical phenomena. The calculators and computers, when programmed with simple software, are then capable of graphing those measurements over time, presenting them in tabular form, or manipulating them in other ways. These opportunities for scientific data collection and analysis are unlike any that have been available to math and science teachers in the past and hold great promise for some true integration of the two disciplines.

IN SUMMARY, measurement offers us the challenge to actively and physically involve children in their learning as well as the opportunity to tie together seemingly diverse components of their mathematics curriculum like fractions and geometry. It also serves as one of the major vehicles by which we can bring the real worlds of the natural and social sciences, health, and physical education into the mathematics classroom.

The study of measurement also provides opportunities for students to further develop their number sense and to practice their counting skills. Only by using measures can students recognize that numbers are often used to describe and compare the properties of physical objects. Students in the early grades should make estimates not only of discrete objects like marbles or seeds but also of continuous properties like the length of a jumprope or the number of children's feet in a dinosaur's footprint.

Students need to focus on identifying the property that they wish to measure. Students need to understand what is meant by the length of an object or its weight or its capacity. Concrete experiences in describing the properties of objects, in sorting objects, and in comparing and contrasting objects provide them with opportunities to develop these concepts.

Students also need experience in making direct comparisons. Which string is longer? Which child is taller? Which rock is heavier? Which glass holds more? Making comparisons will help children better to understand the properties which they are discussing. They also begin to make some indirect measurements. For example, in order to compare the height of the blackboard with the height of a window, they might measure both objects using links and then compare the number of links used for each. Students also use thermometers to measure temperature indirectly, reading the height of the column of mercury to determine how warm or cold it is.

In grades K-2, students should use a variety of non-standard units to measure objects. How many links long is a desk? How many erasers high are you? How many pennies balance a Unifix cube? In each case, students should first be asked to make an estimate and then proceed to actually measure the object. Students should also use different units to measure the same object. They should begin to understand that when the size of a measuring unit increases, the number of units needed to measure the object decreases.

In these grades, students also begin to use standard measurement units in an informal way. Students explore length using inches, feet, centimeters and meters; liquid capacity using quarts, pints, cups, and liters; mass/weight using pounds, ounces, grams, and kilograms; time using days, weeks, months, years, seconds, minutes, and hours; and temperature using degrees Fahrenheit and Celsius.

Whether making direct comparisons, using non-standard units, or using standard measurement units, students in the early grades should always estimate a measure first and then perform the measurement. In this way, their estimation and number sense skills will be reinforced.

Experiences will be such that all students in grades K-2:

A. use and describe measures of length, distance, capacity, weight, area, volume, time, and temperature.

• Students find out how many cubes long their hand is. The class might then generate a graph showing the results.
• Using a large map, students estimate and then use links to find who goes farther to school. This type of activity might be related to a specific story that has been read in class.
• Students name objects big enough to hold a football or too small to hold a soccer ball.
• Students lay out a model zoo with several toy animals, using boxes of different sizes for their cages or yards. They also cut doors of appropriate sizes in the boxes for the animals.
• Students move thermometers to different parts of the room, watching to see how the temperature changes. For example, it may be very hot by the radiators and very cold next to the windows.
• Students name objects they can lift and ones that they cannot lift.
• Students estimate and then use balances to find out how many pennies balance an object.
• Students cut strips of paper to fit around a pumpkin or to make Santa's belt.
• Students fill a large bottle with water using first a juice glass and then a cup. They then compare the measures.
• Students make their own measuring jug using an empty mayonnaise jar. They pour in one cupful of water and mark the water level on the jar with a marker, then another and another.
• Students make their own ruler, marking off intervals equal to the length of one paperclip.
• Students estimate and measure the distance around an object using Unifix cubes or paper clips.
• Students conduct experiments using timers: how many times can you bounce a ball before all the water runs out of the can? how many times can you clap your hands before the sand runs out of the timer? how many times can you blink your eyes in one minute?
• Students make a book describing their day at school. On each page, they stamp a clock face and write underneath a time that the teacher has written on the board. They then draw the hands on the face to show the time. When the actual time of day on the classroom clock matches a time in their book, students draw a picture of what they are doing next to the correct clock face.
• Students line up pattern blocks in different ways to measure the width of a sheet of paper.
• Students build a variety of structures using a specified number of wooden cubes.

• Students compare the lengths of pencils to find out which is longest. They might also be asked to put a set of pencils in order from longest to shortest.
• Students use water, rice, or sand to fill different objects, pouring from one object into another to find out which object holds more and why.
• Students line up in order, from tallest to shortest.
• Children make stick drawings of a family: father, mother, school-aged child, and baby. Which is tallest? Which is shortest?
• Students make boxes out of cardboard and arrange them from smallest to largest.
• Each group of students is given a cup and several containers of different sizes, plain white paper, and 1 inch graph paper. They find out how many cups there are in the container and show the number of cups on the graph paper under a picture of the container. After completing the graph, they put the containers in order from largest to smallest.
• Students compare pattern blocks to see which is taller.

• Students measure the width of their desk using their hands and compare their results. They discuss what would happen if the teacher measured their desks with her hand.
• Students read and discuss How Big Is a Foot? by Rolf Myllar. The king wishes to give the queen a special bed for her birthday and measures the size using his foot. He gives the measurements to the carpenter, who gives them to the little apprentice. The bed is too small, but the apprentice solves the problem and everyone lives happily ever after. They use their own feet to measure the length of the hallway and compare their results. Finally, they measure the hallway using metersticks.

• Students identify things on their body that are the same length as a unit cube from a base tens block set (1 centimeter).
• Students make a list of things that come in quarts and things that come in liters.
• Students play with quart containers to see what they can find out about quarts. They make a list of their findings, including some disagreements. They then proceed to discuss what makes a quart a quart and why their findings disagree.
• Students find out that ten pennies weigh about an ounce.
• Students find that most first-graders are more than a meter tall.

• Students decide whether they should use paper clips or pennies to measure the weight of a pencil.
• Students discuss whether they should use links or metersticks to measure the length of the gym.
• Students write about how they might measure the distance from the cafeteria to their classroom.

• Students estimate how many of their shoes will fit in a giant's footprint and write their estimates. They trace around their shoes and cut them out. After the teacher has pasted a few shoes onto the giant's shoe, the students revise their estimate. They then check the accuracy of their estimates by pasting as many shoes as will fit in the giant's shoe.
• Students estimate the weight of various objects in beans and then use a balance scale to check the accuracy of their measurements.

• Students read The Little Gingerbread Man and make a gingerbread village. In doing so, they measure lengths and capacities.
• Students measure the heights of bean plants at regular intervals, making a graph of their findings.
• 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.

Measurement also help students make connections within mathematics. For example, as students begin to develop understanding of fraction concepts, they extend their understanding of measurement to include fractions of units as well. Measurement is inextricably interwoven with developing understanding of the geometric concepts of perimeter, area, and volume. Furthermore, student develop their estimation skills as they develop their understanding of measurement.

Students continue to develop their ability to identify the property to be measured. In addition to length, distance, capacity, weight, area, volume, time, and temperature, they now are able to discuss the size of angles and the speed of a car or a biker. Students begin to make more indirect measurements. For example, they will measure a desk to find out whether it will fit through a door, or measure how far a toy car goes in a minute and divide to find its speed.

The emphasis in these grades is on moving from non-standard units to the use of standard units of measurement. They solidify their understanding of the basic units introduced in the earlier grades and begin to use fractional units. Students use half-inches, quarter-inches, and eighths of an inch, for example, in measuring the lengths of objects. Students also begin to use some of the larger measures: miles, kilometers, and tons.

Some students may also begin to develop formulas to help count units. For example, students may use shortcuts to find out how many squares cover a rectangle, multiplying the number of rows times the number of squares in each row. Or they may find the distance around an object by measuring each side and then adding the measures.

In summary, in grades 3 and 4, it is important that all students get extensive hands-on experience with measuring properties of a wide variety of physical objects. They will learn to measure by actually doing so with an appropriate measuring instrument.

Building upon K-2 expectations, experiences in grades 3-4 will be such that all students:

A. use and describe measures of length, distance, capacity, weight, area, volume, time, and temperature.

• Students find out how many inches long their hand is. The class might then generate a graph showing the results.
• Using a large map, students estimate and then use centimeter cubes to find who goes farther to school. This type of activity might be related to a specific story that has been read in class.
• Students use rulers to measure the length of the room in feet and inches.
• Students move thermometers to different parts of the school, recording the temperature at each location. For example, it may be very hot in the cafeteria and very cold in the gym. They learn to identify appropriate reference points on both Celsius and Fahrenheit scales (e.g., 30 degrees C is a hot day).
• Students investigate truth-in-packaging by estimating and then using balances to weigh foods.
• Students investigate how many cups in a pint, how many pints in a quart, and how many quarts in a gallon by making lemonade and filling various sizes of containers.
• Students make their own measuring jug using an empty mayonnaise jar. They pour in one cupful of water and mark the water level on the jar with a marker, then another and another.
• Students make their own ruler, marking off intervals equal to the length of one centimeter.
• Students estimate and measure the distance around an object using centimeter cubes or measuring tapes.
• Students conduct experiments using timers: how many times can you bounce a ball, clap your hands, or blink your eyes in one minute?
• Students use calculators to help them find out how many days old they are.
• When going on a field trip, students determine how much time they will have available at a museum by considering when they will arrive and when they must leave.
• Students use wooden cubes to fill rectangular boxes of various sizes as they develop the concept of volume.

• Students compare the areas of different leaves. They use a variety of strategies; some students cover the leaves with centimeter cubes, others make a copy of the leaf on grid paper, and still others just "eyeball" it. They discuss the different strategies used, comparing their advantages and disadvantages.
• Students bring in cereal boxes from home and put them in order from smallest volume to largest volume.
• Students build bridges using straws and pipe cleaners, estimate how many washers their bridge will hold, and then place washers on their bridge until it buckles or breaks. They compare different types of bridges to determine what type is strongest.
• Students estimate and then weigh objects, putting them in order from heaviest to lightest.

• Students measure the length of their classroom using their paces and compare their results. They discuss what would happen if the teacher measured the room with her pace.
• Students read and discuss How Big Is a Foot? by Rolf Myllar. The king wishes to give the queen a special bed for her birthday and measures the size using his foot. He gives the measurements to the carpenter, who gives them to the little apprentice. The bed is too small, but the apprentice solves the problem and everyone lives happily ever after. They use their own feet to measure the length of the hallway and compare their results. Finally, they measure the hallway using metersticks.

• Students identify things on their body that are the same length as one centimeter and use them to measure the length of their pencil.
• Students find out that ten pennies weigh about an ounce.
• Students measure the length of their pace in inches and use that information, along with a measurement of the length of the room in paces, to find the length of the room in inches.

• Students decide what units they should use to measure the weight of a pencil.
• Students discuss what units they should use to measure the length of the gym.
• Students write about how they might measure the distance from the cafeteria to their classroom.

• Students estimate the weight of various objects in grams and then use a balance scale to check the accuracy of their measurements.
• Students estimate the weight and then weigh animal crackers (without the box), graphing their results and comparing their results to the weights indicated on the packages.

• Students measure the heights of bean plants at regular intervals, making a graph of their findings.
• Students estimate and weigh cupfuls of 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 foods to a standard 15 grams, then measuring their weights again the next day.
• Students estimate what fraction of an orange is edible, then weigh oranges, peel them and separate the edible parts, weigh the edible part, and then compute what fraction is actually edible and compare that fraction to their estimate.

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: 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, as students encounter both very small and very large measurement units (such as milligrams or tons), 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. Teachers must take responsibility for furnishing hands-on opportunities that reinforce measurement concepts with all common measures.

The development of measurement formulas is an important part of middle grade mathematics, but being exposed to formulas in the abstract arena of a textbook does not promote understanding. Formulas should be the result of exploration and discovery; they should be developed as an appropriate means of counting iterated units. Less than half of the seventh grade students tested could figure out the area of a rectangle drawn on a sheet of graph paper, and only a little more than half could compute the area given the dimensions of length and height. Often length and width are taught as separate entities and the overlapping of the two to form the square units of area is not emphasized in instruction. In addition, area and perimeter are often confused by middle grade students. Limiting experience to the printed page restricts flexibility so that understanding cannot be generalized.

In order to further strengthen students' understanding of measurement concepts, it is important to connect measurement to other ideas in mathematics. Students should manipulate objects (e.g., measure), represent the information gathered visually (e.g., graph), model the situation with symbols (e.g., 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 generate 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 must use measuring instruments until that use becomes second nature. The curriculum should focus on the development of understanding of measurement rather than on the rote memorization of formulas. This can be reinforced by teaching students to estimate and to be aware of context in their estimates, such as estimating too high when buying carpeting. Students must be given the opportunity to extend their learning to new situations and new applications.

Bright, G. W. & Hoeffner, K. "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., Mead, N. A., & Phillips, G. W. A World of Differences: An International Assessment of Mathematics and Science. Princeton, N. J. : Educational Testing Service, 1989.

Building upon K-4 expectations, experiences in grades 5-6 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 triangle. Then they enclose the triangle in a rectangle and use 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, measuring its length and width, and counting how many squares are covered by the foot.

• Students use a given scale to compute the actual length of a variety of illustrated dinosaurs.
• Students make a scale drawing of their classroom.
• Students use a map to find the distance between two cities.

• 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.

• Students measure a given line and compare their results, focussing 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 could be slightly more or less.

• 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 cubes and looking for patterns in their results. They describe the "shortcut" way to find the volume in their journals.

• 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.

• 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
  20 degrees C is about 70 degrees F (room temperature)

• 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 wood (posterboard) before cutting.
• Students compare the measurements of an object to that of its shadow on a wall as the distance between the object and the wall increases.

• Students use pattern blocks to see how the area of a square changes when the length is doubled. They then repeat the experiment using equilateral triangles.
• Students use cubes to explore how the volume of a cube changes when the length of the side is doubled.
• Students use graph paper to draw as many rectangles as they can that have a perimeter of 16 units. They then find the area of each rectangle, look for patterns, and summarize their results.

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.

Students at the high school level will frequently use measurement to help develop understanding of other mathematical concepts. For example, students may use a computer program that measures angles to help them discover the relationship between the measures of two vertical angles formed by intersecting lines or the measures of inscribed and circumscribed angles intercepting the same arc of a circle. They may also develop algebraic techniques that help them to find measures, as, for example, when they develop a formula for finding the distance between two points in the coordinate plane.

High school students also use measurement frequently in connection with other subject areas. Science experiments generally require some use of measurement. Social studies activities often require students to read and interpret maps and/or scale drawings. In technology classes, woodshop, drafting, sewing, and cooking, students must also use a variety of measuring tools and techniques. Even in physical education, students frequently will measure distances (approximately or exactly) and rates.

Building upon K-8 expectations, experiences in grades 9-12 will be such that all students:

Q. apply their knowledge of measurement in 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 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 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 build scale models of the classroom, the school, or a monument.

• Students use significant digits appropriately in measuring large distances, such as the distance from one school to another.
• Students explain how accurate measurements of distances on maps are by referring to the degree of accuracy of a measurement.
• In making a scale drawing of their bedroom, students discuss the degree of accuracy of their measurements.

• Students use coordinate geometry techniques to determine the distance between two points.
• Students use similar figures and proportions to measure the height of a tree or a flagpole.
• Students use the Pythagorean Theorem to determine how long a ladder is needed to climb a wall.
• Students use right-triangle trigonometry to measure the width of a canyon or the height of a waterfall.

• Students investigate how the volume of a cereal box changes with its area by finding the volume and surface area of a box of their favorite kind of cereal. They also discuss how the shape of the box affects its volume and surface area and why the volume of the box is so large for the amount of cereal it contains.
• Students measure their own femurs and heights to determine the relationship between the length of the femur and the height of a person.

• Students use computer drawing and measuring utilities to discover geometric concepts. They also discuss the limitations of such a program. For example, a program may give 14.7 and 7.3 for the lengths of the base and the midline of a triangle because of its measurement limitations.
• Students determine what kind of measuring instrument needs to be used to measure ingredients for pain-relievers, for cough syrup, for a cake, and for a stew. They discuss the accuracy and error of each measurement.