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Measurement is a process that students in grades 3–5 use every day as they explore questions related to their school or home environment. For example, how much catsup is used in the school cafeteria each day? What is the distance from my house to the school? What is the range of heights of players on the basketball team? Such questions require students to use concepts and tools of measurement to collect data and to describe and quantify their world. In grades 3–5, measurement helps connect ideas within areas of mathematics and between mathematics and other disciplines. It can serve as a context to help students understand important mathematical concepts such as fractions, geometric shapes, and ways of describing data.
Prior to grade 3, students should have begun to develop an understanding of what
it means to measure an object, that is, identifying an attribute to be
measured, choosing an appropriate unit, and comparing that unit to the
object being measured. They should have had many experiences with measuring
length and should also have explored ways to measure liquid volume, weight,
and time. In grades 3–5, students should deepen and expand their
understanding and use of measurement. For example, they should measure
other attributes such as area and angle. They need to begin paying closer
attention to the degree of accuracy when measuring and use a wider variety
of measurement tools. They should also begin to develop and use formulas
for the measurement of certain attributes, such as area.
In learning about measurement and learning how to measure, students should be
actively involved, drawing on familiar and accessible contexts. For example,
students in grades 3–5 should measure objects and space in their
classroom or use maps to determine locations and distances around their
community. They should determine an appropriate unit and use it to measure
the area of their classroom's floor, estimate the time it takes to do
various tasks, and measure and represent change in the size of attributes,
such as their height. »
Students in grades 3–5 should measure the attributes of a variety of physical objects and extend their work to measuring more complex attributes, including area, volume, and angle. They will learn that length measurements in particular contexts are given specific names, such as perimeter, width, height, circumference, and distance. They can begin to establish some benchmarks by which to estimate or judge the size of objects. For example, they learn that a "square corner" is called a right angle and establish this as a benchmark for estimating the size of other angles.
Students in grades 3–5 should be able to recognize the need to select units appropriate to the attribute being measured. Different kinds of units are needed for measuring area than for measuring length. At first they might use convenient nonstandard units such as lima beans to estimate area and then come to recognize the need for a standard unit such as a unit square. Likewise, the need for a standard three-dimensional unit to measure volume grows out of initial experiences filling containers with items such as rice or packing pieces. As students find that there are spaces between the units, that the units are not easy to count, or that the units are not of a uniform size, they will appreciate the need for a standard unit.
In these grades, more emphasis should be placed on the standard units that are used to communicate in the United States (the customary units) and around the world (the metric system). Students should become familiar with the common units in these systems and establish mental images or benchmarks for judging and comparing size. For example, they may know that a paper clip weighs about a gram, the width of their forefinger is about a centimeter, or the distance from their elbow to their fingertip is about a foot.
Students should gain facility in expressing measurements in equivalent forms. They use their knowledge of relationships between units and their understanding of multiplicative situations to make conversions, such as expressing 150 centimeters as 1.5 meters or 3 feet as 36 inches. Since students in the United States encounter two systems of measurement, they should also have convenient referents for comparing units in different systems—for example, 2 centimeters is a little less than an inch, a quart is a little less than a liter, a kilogram is about two pounds. However, they do not need to make formal conversions between the two systems at this level.
Students in grades 3–5 should encounter the notion that measurements in
the real world are approximate, in part because of the instruments used
and because of human error in reading the scales of these instruments.
For example, figure 5.17 describes a measurement task and summarizes results
typical of what groups of students obtain. Such an exercise provides a
context in which the teacher can raise, and the class can consider, the
idea of measurement as an estimation process.
Each pair of students will find slightly different measurements, even though they are measuring the same object using the same kind of measurement tools. The teacher should ask students to discuss the factors that may lead to different measurements. Students' responses will vary according to their experience, but by grade 5 they should recognize factors that affect precision. These include the limitations of the measurement tool, how precisely students read the scale on the measuring » instrument (was the scale marked and read in centimeters or millimeters?), and the students' perceived need for accuracy. The discussion might lead to considering the importance of measuring precisely in certain contexts. For instance, carpenters often measure twice and use special instruments in order to minimize the waste of materials, but an estimate might be quite adequate in other instances (e.g., the scout troop hiked about 2.5 miles).
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In grades 3–5, an expanded number of tools and range of measurement techniques should be available to students. When using conventional tools such as rulers and tape measures for measuring length, students will need instruction to learn to use these tools properly. For example, they will need to recognize and understand the markings on a ruler, including where the "0," or beginning point, is located. When standard measurement tools are difficult to use in a particular situation, they must learn to adapt their tools or invent techniques that will work. In the earlier example (fig. 5.17) measuring the circumference of a clock face with a rigid ruler presented a particular challenge. Using string or some other flexible object to outline the clock face and then measuring the string would have been a good strategy. Students should be challenged to develop measurement techniques as needed in order to measure complex figures or objects. For example, they might measure » the area of an irregular polygon or a leaf by covering it with transparent grid paper and counting units or by breaking it apart into regular shapes that they can measure.
Students in grades 3–5 should develop strategies to estimate measurements. For example, to estimate the length of the classroom, they might estimate the length of one floor tile and then count the number of tiles across the room and multiply the length by the number of tiles. Another strategy for estimating measurements is to compare the item to be measured against some benchmark. For example, a student might estimate the teacher's height by noting that it is about one and a quarter times the student's own height. This particular strategy highlights the use of multiplicative reasoning, an important indication of advancing understanding.
Strategies for estimating measurements
are varied and often depend on the particular situation. By sharing strategies,
students can compare and evaluate different approaches. Students also
need experience in judging what degree of accuracy is required in a given
situation and whether an underestimate or overestimate is more desirable.
For example, in estimating the time needed to get up in the morning, eat
breakfast, and walk or drive to school, an overestimate makes sense. However,
an underestimate of the time needed to cook vegetables on the grill might
be considered appropriate, since more time can always be added to the
cooking process but not taken away from it.
As students have opportunities to look for patterns in the results of their measurements,
they recognize that their methods for measuring the area and volume of
particular objects can be generalized as formulas. For example, the table
in figure 5.19 is typical of what groups of third graders might produce
when using a transparent grid to determine »
the areas of a set of rectangles. As they begin generating the
table, they realize that counting all the squares is not necessary once
the length (L) and width (W) of the rectangle are
determined with the grid. They test their conjecture that Area = L
W, and
it appears to work for each rectangle in the set. Later, their teacher
challenges them to think about whether and why their formula will work
for big rectangles as well as small ones.
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Students in grades 3–5 should develop strategies for determining surface area and volume on the basis of concrete experiences. They should measure various rectangular solids using objects such as tiles and cubes, organize the information, look for patterns, and then make generalizations. For example, the "tower of cubes" problem in figure 5.5 highlights the kind of activity that builds from concrete experiences and leads to generalizations, including the development of general formulas for measuring surface area and volume. These concrete experiences are essential in helping students understand the relationship between the measurement of an object and the succinct formula that produces the measurement.
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