Home > Progressions for the Common Core State Standards in Mathematics (draft)

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Overview

Geometric measurement connects the two most critical domains of early mathematics, geometry and number, with each providing con- ceptual support to the other. Measurement is central to mathematics, to other areas of mathematics (e.g., laying a sensory and concep- tual foundation for arithmetic with fractions), to other subject matter domains, especially science, and to activities in everyday life. For these reasons, measurement is a core component of the mathematics curriculum. Measurement is the process of assigning a number to a mag- nitude of some attribute shared by some class of objects, such as length, relative to a unit. Length is a

Technically, mass is the amount of matter in an object. Weight is the force exerted on the body by gravity. On the earth��s surface, the distinction is not important (on the moon, an object would have the same mass, would weight less due to the lower gravity).

Before learning to measure attributes, children need to recog- nize them, distinguishing them from other attributes. That is, the attribute to be measured has to ��stand out�� for the student and be discriminated from the undifferentiated sense of amount that young children often have, labeling greater lengths, areas, volumes, and so forth, as ��big�� or ��bigger.�� Students then can become increasingly competent at

1This progression concerns Measurement and Data standards related to geomet-

ric measurement. The remaining Measurement and Data standards are discussed in the K–3 Categorical Data and Grades 2–5 Measurement Data Progressions.

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Measurement stated in the Standards as: If the length of object A is greater than the length of object B, and the length of object B is greater than the length of object C, then the length of ob- ject A is greater than the length of object C. This principle applies to measurement of other quanti- ties as well. Students should apply the principle of transitivity of measurement to make indirect comparisons, but they need not use this techni- cal term.

The purpose of measurement is to allow indirect comparisons of objects�� amount of an attribute using numbers. An attribute of an object is measured (i.e., assigned a number) by comparing it to an amount of that attribute held by another object. One measures length with length, mass with mass, torque with torque, and so on. In geometric measurement, a unit is chosen and the object is subdivided or partitioned by copies of that unit and, to the necessary degree of precision, units subordinate to the chosen unit, to determine the number of units and subordinate units in the partition. Personal benchmarks, such as ��tall as a doorway�� build students�� intuitions for amounts of a quantity and help them use measurements to solve practical problems. A combination of internalized units and measurement processes allows students to develop increasing ac- curate estimation competencies. Both in measurement and in estimation, the concept of

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Grade 3 Grade 5

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Kindergarten

Describe and compare measurable attributes Students often ini- tially hold undifferentiated views of measurable attributes, saying that one object is ��bigger�� than another whether it is longer, or greater in area, or greater in volume, and so forth. For example, two students might both claim their block building is ��the biggest.�� Conversations about how they are comparing—one building may be taller (greater in length) and another may have a larger base (greater in area)—help students learn to discriminate and name these measureable attributes. As they discuss these situations and compare objects using different attributes, they learn to distinguish, label, and describe several measureable attributes of a single object.K.MD.1

K.MD.1Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single ob- ject.

Thus, teachers listen for and extend conversations about things that are ��big,�� or ��small,�� as well as ��long,�� ��tall,�� or ��high,�� and name, dis- cuss, and demonstrate with gestures the attribute being discussed (length as extension in one dimension is most common, but area, volume, or even weight in others).

K.MD.2Directly compare two objects with a measurable attribute in common, to see which object has ��more of��/��less of�� the at- tribute, and describe the difference.

compare only one endpoint of objects to say which is longer. Dis-

cussing such situations (e.g., when a child claims that he is ��tallest�� because he is standing on a chair) can help students resolve and coordinate perceptual and conceptual information when it conflicts. Teachers can reinforce these understandings, for example, by hold- ing two pencils in their hand showing only one end of each, with the longer pencil protruding less. After asking if they can tell which pencil is longer, they reveal the pencils and discuss whether children were ��fooled.�� The necessity of aligning endpoints can be explicitly addressed and then re-introduced in the many situations throughout the day that call for such comparisons. Students can also make such comparisons by moving shapes together to see which has a longer side. Even when students seem to understand length in such activities, they may not conserve length. That is, they may believe that if one of two sticks of equal lengths is vertical, it is then longer than the other, horizontal, stick. Or, they may believe that a string, when bent or curved, is now shorter (due to its endpoints being closer to each other). Both informal and structured experiences, including demon- strations and discussions, can clarify how length is maintained, or conserved, in such situations. For example, teachers and students might rotate shapes to see its sides in different orientations. As with number, learning and using language such as ��It looks longer, but it really isn��t longer�� is helpful. Students who have these competencies can engage in experi- ences that lay the groundwork for later learning. Many can begin

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Grade 1

Length comparisons First graders should continue to use direct comparison—carefully, considering all endpoints—when that is ap- propriate. In situations where direct comparison is not possible or convenient, they should be able to use indirect comparison and ex- planations that draw on transitivity (MP3). Once they can compare lengths of objects by direct comparison, they could compare sev- eral items to a single item, such as finding all the objects in the classroom the same length as (or longer than, or shorter than) their forearm.1.MD.1 Ideas of transitivity can then be discussed as they use

1.MD.1Order three objects by length; compare the lengths of two objects indirectly by using a third object.

a string to represent their forear��s length. As another example, stu- dents can figure out that one path from the teachers�� desk to the door is longer than another because the first path is longer than a length of string laid along the path, but the other path is shorter than that string. Transitivity can then be explicitly discussed: If

1.MD.1Order three objects by length; compare the lengths of two objects indirectly by using a third object.

multiple comparisons. Initially, students find it difficult to seriate a large set of objects (e.g., more than 6 objects) that differ only slightly in length. They tend to order groups of two or three objects, but they cannot correctly combine these groups while putting the objects in order. Completing this task efficiently requires a systematic strat- egy, such as moving each new object ��down the line�� to see where it fits. Students need to understand that each object in a seriation is larger than those that come before it, and shorter than those that come after. Again, reasoning that draws on transitivity is relevant. Such seriation and other processes associated with the mea- surement and data standards are important in themselves, but also play a fundamental role in students�� development. The general rea- soning processes of seriation, conservation (of length and number), and classification (which lies at the heart of the standards discussed in the K–3 Categorical Data Progression) predict success in early childhood as well as later schooling. Measure lengths indirectly and by iterating length units Directly comparing objects, indirectly comparing objects, and ordering ob- jects by length are important practically and mathematically, but they are not length measurement, which involves assigning a num- ber to a length. Students learn to lay physical units such as cen- timeter or inch manipulatives end-to-end and count them to measure a length.1.MD.2 Such a procedure may seem to adults to be straight-

1.MD.2Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps.

forward, however, students may initially iterate a unit leaving gaps between subsequent units or overlapping adjacent units. For such students, measuring may be an activity of placing units along a

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Although, from the adult perspective, the lengths of the rows were the same, many children argued that the row with 6 matches was longer because it had more matches. They counted units (matches), assigning a number to a

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Grade 2

Measure and estimate lengths in standard units Second graders learn to measure length with a variety of tools, such as rulers, me- ter sticks, and measuring tapes.2.MD.1 Although this appears to some

2.MD.1Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

adults to be relatively simple, there are many conceptual and proce- dural issues to address. For example, students may begin counting at the numeral ��1�� on a ruler. The numerals on a ruler may signify to students when to start counting, rather than the amount of space that has already been covered. It is vital that students learn that ��one�� represents the space from the beginning of the ruler to the hash mark, not the hash mark itself. Again, students may not un- derstand that units must be of equal size. They will even measure with tools subdivided into units of different sizes and conclude that quantities with more units are larger. To learn measurement concepts and skills, students might use both simple rulers (e.g., having only whole units such as centimeters or inches) and physical units (e.g., manipulatives that are centimeter or inch lengths). As described for Grade 1, teachers and students can call these ��length-units.�� Initially, students lay multiple copies of the same physical unit end-to-end along the ruler. They can also progress to iterating with one physical unit (i.e., repeatedly marking off its endpoint, then moving it to the next position), even though this is more difficult physically and conceptually. To help them make the transition to this more sophisticated understanding of measurement, students might draw length unit marks along sides of geometric shapes or other lengths to see the unit lengths. As they measure with these tools, students with the help of the teacher discuss the concepts and skills involved, such as the following.

K.CC.4Understand the relationship between numbers and quan- tities; connect counting to cardinality.

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should be simply ways to help count the iteration of length-units. Frequently comparing results of measuring the same object with manipulative standard units and with these rulers helps students connect their experiences and ideas. As they build and use these tools, they develop the ideas of length-unit iteration, correct align- ment (with a ruler), and the zero-point concept (the idea that the zero of the ruler indicates one endpoint of a length). These are re- inforced as children compare the results of measuring to compare to objects with the results of directly comparing these objects. Similarly, discussions might frequently focus on ��

2.MD.2Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

will take more centimeter lengths to cover a certain distance than inch lengths because inches are the larger unit. Initially, students may not appreciate the need for identical units. Previously described work with manipulative units of standard measure (e.g., 1 inch or 1 cm), along with related use of rulers and consistent discussion, will help children learn both the concepts and procedures of linear mea- surement. Thus, second grade students can learn that the larger the unit, the fewer number of units in a given measurement (as was illustrated on p. 9). That is, for measurements of a given length there is an inverse relationship between the size of the unit of measure and the number of those units. This is the time that measuring and reflecting on measuring the same object with different units, both standard and nonstandard, is likely to be most productive (see the discussion of this issue in the Grade 1 section on length). Results of measuring with different nonstandard length-units can be explic- itly compared. Students also can use the concept of unit to make

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2.MD.4Measure to determine how much longer one object is than another, expressing the length difference in terms of a stan- dard length unit. 2.MD.5Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

a simple unit ruler or put a length of connecting cubes together to measure first one modeling clay ��snake,�� then another, to find the total of their lengths. The snakes can be laid along a line, allowing students to compare the measurement of that length with the sum of the two measurements. Second graders also begin to apply the concept of length in less obvious cases, such as the width of a circle, the length and width of a rectangle, the diagonal of a quadrilateral, or the height of a pyramid. As an arithmetic example,

students might measure all the sides of a table with unmarked (foot) rulers to measure how much ribbon they would need to decorate the perimeter of the table.2.MD.5 They learn to measure two objects and subtract the smaller measurement from the larger to find how much longer one object is than the other. Second graders can also learn to represent and solve numer- ical problems about length using tape or number-bond diagrams. (See p. 16 of the Operations and Algebraic Thinking Progression for discussion of when and how these diagrams are used in Grade 1.) Students might solve two-step numerical problems at different levels of sophistication (see p. 18 of the Operations and Algebraic Thinking Progression for similar two-step problems involving dis- crete objects). Conversely, ��missing measurements�� problems about length may be presented with diagrams.

What are the missing lengths of the third and fourth sides of the rectangle?

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Assume all seg- ments in ��steps�� are the same length What are the missing lengths of each step and the bottom of the stairway?

These problems might be presented in the context of turtle geometry. Students work on paper to figure out how far the Logo turtle would have to travel to finish drawing the house (the remainder of the right side, and the bottom). They then type in Logo commands (e.g., for the rectangle, forward 40 right 90 fd 100 rt 90 fd 20 fd 20 rt 90 fd 100) to check their calculations (MP5).

These understandings are essential in supporting work with num- ber line diagrams.2.MD.6 That is, to use a number line diagram to

2.MD.6Represent whole numbers as lengths from 0 on a num- ber line diagram with equally spaced points corresponding to the numbers 0, 1, 2, . . . , and represent whole-number sums and dif- ferences within 100 on a number line diagram.

understand number and number operations, students need to un- derstand that number line diagrams have specific conventions: the use of a single position to represent a whole number and the use of marks to indicate those positions. They need to understand that a number line diagram is like a ruler in that consecutive whole numbers are 1 unit apart, thus they need to consider the distances between positions and segments when identifying missing numbers. These understandings underlie students�� successful use of number line diagrams. Students think of a number line diagram as a mea- surement model and use strategies relating to distance, proximity of numbers, and reference points. After experience with measuring, second graders learn to esti- mate lengths.2.MD.3 Real-world applications of length often involve

2.MD.3Estimate lengths using units of inches, feet, centimeters, and meters.

estimation. Skilled estimators move fluently back and forth between written or verbal length measurements and representations of their corresponding magnitudes on a

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2.G.2Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

and this work is an ideal context in which to simultaneously develop both arithmetical and spatial structuring foundations for later work with area.

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Grade 3

Perimeter Third graders focus on solving real-world and mathe- matical problems involving perimeters of polygons.3.MD.8 A perime-

3.MD.8Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rect- angles with the same perimeter and different areas or with the same area and different perimeters.

ter is the boundary of a two-dimensional shape. For a polygon, the length of the perimeter is the sum of the lengths of the sides. Initially, it is useful to have sides marked with unit length marks, allowing students to count the unit lengths. Later, the lengths of the sides can be labeled with numerals. As with all length tasks, stu- dents need to count the length-units and not the end-points. Next, students learn to mark off unit lengths with a ruler and label the length of each side of the polygon. For rectangles, parallelograms, and regular polygons, students can discuss and justify faster ways to find the perimeter length than just adding all of the lengths (MP3). Rectangles and parallelograms have opposite sides of equal length, so students can double the lengths of adjacent sides and add those numbers or add lengths of two adjacent sides and double that num- ber. A regular polygon has all sides of equal length, so its perimeter length is the product of one side length and the number of sides.

The perimeter of this rectangle is 168 length units. What are the lengths of the three unla- beled sides?

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Assume all short segments are the same length and all angles are right

Compare these problems with the ��missing measurements�� problems of Grade 2. Another type of perimeter problem is to draw a robot on squared grid paper that meets specific criteria. All the robot��s body parts must be rectangles. The perimeter of the head might be 36 length-units, the body, 72; each arm, 24; and each leg, 72. Students are asked to provide a convincing argument that their robots meet these criteria (MP3). Next, students are asked to figure out the area of each of their body parts (in square units). These are discussed, with students led to reflect on the different areas that may be produced with rectangles of the same perimeter. These types of problems can be also presented as turtle geometry problems. Students create the commands on paper and then give their commands to the Logo turtle to check their calculations. For turtle length units, the perimeter of the head might be 300 length-units, the body, 600; each arm, 400; and each leg, 640.

Perimeter problems for rectangles and parallelograms often give only the lengths of two adjacent sides or only show numbers for these sides in a drawing of the shape. The common error is to add just those two numbers. Having students first label the lengths of the other two sides as a reminder is helpful. Students then find unknown side lengths in more difficult ��miss- ing measurements�� problems and other types of perimeter prob- lems.3.MD.8 Children learn to subdivide length-units. Making one��s own ruler and marking halves and other partitions of the unit may be helpful in this regard. For example, children could fold a unit in halves, mark the fold as a half, and then continue to do so, to build fourths and eighths, discussing issues that arise. Such activities relate to fractions on the number line.3.NF.2 Labeling all of the fractions can

3.NF.2Understand a fraction as a number on the number line; represent fractions on a number line diagram.

help students understand rulers marked with halves and fourths but not labeled with these fractions. Students also measure lengths using rulers marked with halves and fourths of an inch.3.MD.4 They

3.MD.4Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units—whole numbers, halves, or quarters.

show these data by making a line plot, where the horizontal scale is marked off in appropriate units—whole numbers, halves, or quarters (see the Measurement Data Progression, p. 10). Understand concepts of area and relate area to multiplication and to addition Third graders focus on learning area. Students learn formulas to compute area, with those formulas based on, and sum- marizing, a firm conceptual foundation about what area is. Stu- dents need to learn to conceptualize area as the amount of two- dimensional space in a bounded region and to measure it by choos- ing a unit of area, often a square. A two-dimensional geometric figure that is covered by a certain number of squares without gaps

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3.MD.5Recognize area as an attribute of plane figures and un- derstand concepts of area measurement.

Activities such as those in the Geometry Progression teach stu- dents to compose and decompose geometric regions. To begin an ex- plicit focus on area, teachers might then ask students which of three rectangles covers the most area. Students may first solve the prob-

lem with decomposition (cutting and/or folding) and re-composition, and eventually analyses with area-units, by covering each with unit squares (tiles).3.MD.5, 3.MD.6 Discussions should clearly distinguish

3.MD.5Recognize area as an attribute of plane figures and un- derstand concepts of area measurement. 3.MD.6Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units).

the attribute of area from other attributes, notably length. Students might then find the areas of other rectangles. As pre- viously stated, students can be taught to multiply length measure- ments to find the area of a rectangular region. But, in order that they make sense of these quantities (MP2), they first learn to in- terpret measurement of rectangular regions as a multiplicative re- lationship of the number of square units in a row and the number of rows.3.MD.7a This relies on the development of spatial structuring.MP7

3.MD.7aFind the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.

To build from spatial structuring to understanding the number of

MP7 See the Geometry Progression

area-units as the product of number of units in a row and number of rows, students might draw rectangular arrays of squares and learn to determine the number of squares in each row with increasingly sophisticated strategies, such as skip-counting the number in each row and eventually multiplying the number in each row by the num- ber of rows (MP8). They learn to partition a rectangle into identical squares by anticipating the final structure and forming the array by drawing line segments to form rows and columns. They use skip counting and multiplication to determine the number of squares in the array. Many activities that involve seeing and making arrays of squares to form a rectangle might be needed to build robust conceptions of a rectangular area structured into squares. One such activity is illustrated in the margin. In this progression, less sophisticated

activities of this sort were suggested for earlier grades so that Grade 3 students begin with some experience. Students learn to understand and explain why multiplying the side lengths of a rectangle yields the same measurement of area as counting the number of tiles (with the same unit length) that fill the rectangle��s interior (MP3).3.MD.7a For example, students might explain that one length tells how many unit squares in a row and the other length tells how many rows there are. Students might then solve numerous problems that involve rect- angles of different dimensions (e.g., designing a house with rooms that fit specific area criteria) to practice using multiplication to com- pute areas.3.MD.7b The areas involved should not all be rectangular,

3.MD.7bMultiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number prod- ucts as rectangular areas in mathematical reasoning.

but decomposable into rectangles (e.g., an ��L-shaped�� room).3.MD.7d

3.MD.7dRecognize area as additive. Find areas of rectilinear fig- ures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this tech- nique to solve real world problems.

Students also might solve problems such as finding all the rect- angular regions with whole-number side lengths that have an area of 12 area-units, doing this later for larger rectangles (e.g., enclosing 24, 48, or 72 area-units), making sketches rather than drawing each

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3.MD.7cUse tiling to show in a concrete case that the area of a rectangle with whole-number side lengths

that the area of a rectangular region of, for example, 12 length-units by 5 length-units can be found either by multiplying 12 5, or by adding two products, e.g., 10 5 and 2 5, illustrating the distributive property. Recognize perimeter as an attribute of plane figures and distin- guish between linear and area measures With strong and distinct concepts of both perimeter and area established, students can work on problems to differentiate their measures. For example, they can find and sketch rectangles with the same perimeter and different ar- eas or with the same area and different perimeters and justify their claims (MP3).3.MD.8 Differentiating perimeter from area is facilitated

3.MD.8Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rect- angles with the same perimeter and different areas or with the same area and different perimeters.

by having students draw congruent rectangles and measure, mark off, and label the unit lengths all around the perimeter on one rect- angle, then do the same on the other rectangle but also draw the square units. This enables students to see the units involved in length and area and find patterns in finding the lengths and areas of non-square and square rectangles (MP7). Students can continue to describe and show the units involved in perimeter and area after they no longer need these . Problem solving involving measurement and estimation of inter- vals of time, liquid volumes, and masses of objects Students in Grade 3 learn to solve a variety of problems involving measurement and such attributes as length and area, liquid volume, mass, and time.3.MD.1, 3.MD.2 Many such problems support the Grade 3 empha-

3.MD.1Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram. 3.MD.2Measure and estimate liquid volumes and masses of ob- jects using standard units of grams (g), kilograms (kg), and liters (l).2 Add, subtract, multiply, or divide to solve one-step word prob- lems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measure- ment scale) to represent the problem.3

sis on multiplication (see Table 1) and the mathematical practices of making sense of problems (MP1) and representing them with equa- tions, drawings, or diagrams (MP4). Such work will involve units of mass such as the kilogram.

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Unknown Product Group Size Unknown Number of Groups Unknown A B A C and C A B C and C B

A few words on volume are relevant. Compared to the work in area, volume introduces more complexity, not only in adding a third dimension and thus presenting a significant challenge to students�� spatial structuring, but also in the materials whose volumes are measured. These materials may be solid or fluid, so their volumes are generally measured with one of two methods, e.g., ��packing�� a right rectangular prism with cubic units or ��filling�� a shape such as a right circular cylinder. Liquid measurement, for many third graders, may be limited to a one-dimensional unit structure (i.e., simple iter- ative counting of height that is not processed as three-dimensional). Thus, third graders can learn to measure with liquid volume and to solve problems requiring the use of the four arithmetic opera- tions, when liquid volumes are given in the same units throughout each problem. Because liquid measurement can be represented with one-dimensional scales, problems may be presented with drawings or diagrams, such as measurements on a beaker with a measurement scale in milliliters.

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Grade 4

In Grade 4, students build on competencies in measurement and in building and relating units and units of units that they have devel- oped in number, geometry, and geometric measurement.

4.MD.1Know relative sizes of measurement units within one sys- tem of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table.

Solve problems involving measurement and conversion of mea- surements from a larger unit to a smaller unit Fourth graders learn the relative sizes of measurement units within a system of measurement4.MD.1 including:

Super- or subordinate unit Length in terms of basic unit kilometer 103 or 1000 meters hectometer 102 or 100 meters decameter 101 or 10 meters meter 1 meter decimeter 10 1 or 1

10 meters

centimeter 10 2 or 1

100 meters

millimeter 10 3 or 1

1000 meters

Note the similarity to the structure of base-ten units and U.S. currency (see illustrations on p. 12 of the Number and Operations in Base Ten Progression).

Expressing larger measurements in smaller units within the met- ric system is an opportunity to reinforce notions of place value. There are prefixes for multiples of the basic unit (meter or gram), although only a few (kilo-, centi-, and milli-) are in common use. Tables such as the one in the margin indicate the meanings of the prefixes by showing them in terms of the basic unit (in this case, meters). Such tables are an opportunity to develop or reinforce place value concepts and skills in measurement activities.

4.MD.2Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or deci- mals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measure- ment quantities using diagrams such as number line diagrams that feature a measurement scale.

Relating units within the metric system is another opportunity to think about place value. For example, students might make a table that shows measurements of the same lengths in centimeters and meters. Relating units within the traditional system provides an oppor- tunity to engage in mathematical practices, especially ��look for and make use of structure�� (MP7) and ��look for and express regularity in repeated reasoning�� (MP8). For example, students might make a table that shows measurements of the same lengths in feet and inches.

3 times as much orange as strawberry

In this diagram, quantities are represented on a measurement scale.

Students also combine competencies from different domains as they solve measurement problems using all four arithmetic opera- tions, addition, subtraction, multiplication, and division (see exam- ples in Table 1).4.MD.2 For example, ��How many liters of juice does the class need to have at least 35 cups if each cup takes 225 ml?�� Students may use tape or number line diagrams for solving such problems (MP1).

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Students learn to consider perimeter and area of rectangles, be- gun in Grade 3, more abstractly (MP2). Based on work in previous grades with multiplication, spatially structuring arrays, and area, they abstract the formula for the area of a rectangle

a unit of length, a rectangle whose sides have length

Students generate and discuss advantages and disadvantages of various formulas for the perimeter length of a rectangle that is

2

also helpful. For example, a verbal summary of the basic formula,

3.MD.8Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rect- angles with the same perimeter and different areas or with the same area and different perimeters.

and maintaining the distinction in Grade 4 and later grades, where rectangle perimeter and area problems may get more complex and problem solving can benefit from knowing or being able to rapidly remind oneself of how to find an area or perimeter. By repeatedly reasoning about how to calculate areas and perimeters of rectangles, students can come to see area and perimeter formulas as summaries of all such calculations (MP8).

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4.MD.3Apply the area and perimeter formulas for rectangles in real world and mathematical problems.

ample, they might be asked, ��A rectangular garden has as an area of 80 square feet. It is 5 feet wide. How long is the garden?�� Here, specifying the area and the width, creates an unknown factor prob- lem (see Table 1). Similarly, students could solve perimeter problems that give the perimeter and the length of one side and ask the length of the adjacent side. Students could be challenged to solve multi- step problems such as the following. ��A plan for a house includes rectangular room with an area of 60 square meters and a perimeter of 32 meters. What are the length and the width of the room?�� In Grade 4 and beyond, the mental visual images for perime- ter and area from Grade 3 can support students in problem solving with these concepts. When engaging in the mathematical practice of reasoning abstractly and quantitatively (MP2) in work with area and perimeter, students think of the situation and perhaps make a drawing. Then they recreate the ��formula�� with specific numbers and one unknown number as a situation equation for this particu- lar numerical situation.

structs an equation as a representation of a situation rather than identifying the situation as an example of a familiar equation.

down a memorized formula and put in known values because at Grade 4 students do not evaluate expressions (they begin this type of work in Grade 6). In Grade 4, working with perimeter and area of rectangles is still grounded in specific visualizations and num- bers. These numbers can now be any of the numbers used in Grade 4 (for addition and subtraction for perimeter and for multiplication and division for area).4.NBT.4, 4.NF.3d, 4.OA.4 By repeatedly reasoning

4.NBT.4Fluently add and subtract multi-digit whole numbers us- ing the standard algorithm. 4.NF.3dSolve word problems involving addition and subtraction of fractions referring to the same whole and having like denom- inators, e.g., by using visual fraction models and equations to represent the problem. 4.OA.4Find all factor pairs for a whole number in the range 1– 100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1– 100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1–100 is prime or composite.

about constructing situation equations for perimeter and area involv- ing specific numbers and an unknown number, students will build a foundation for applying area, perimeter, and other formulas by substituting specific values for the variables in later grades. Understand concepts of angle and measure angles Angle mea- sure is a ��turning point�� in the study of geometry. Students often find angles and angle measure to be difficult concepts to learn, but that learning allows them to engage in interesting and important mathematics. An

one to the other about

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360 of a circle

is called a ��one-degree angle,�� and degrees are the unit used to measure angles in elementary school. A full rotation is thus 360 .

Two angles are called

Like length, area, and volume, angle measure is additive: The sum of the measurements of

signed to evoke a dynamic image of turning, such as hinges or doors, many students use the length between the endpoints, thus teachers find it useful to repeatedly discuss such cognitive ��traps.�� As with other concepts (e.g., see the Geometry Progression), stu- dents need varied examples and explicit discussions to avoid learn- ing limited ideas about measuring angles (e.g., misconceptions that a right angle is an angle that points to the right, or two right angles represented with different orientations are not equal in measure). If examples and tasks are not varied, students can develop incomplete and inaccurate notions. For example, some come to associate all slanted lines with 45 measures and horizontal and vertical lines with measures of 90 . Others believe angles can be ��read off�� a

4.MD.6Measure angles in whole-number degrees using a pro- tractor. Sketch angles of specified measure.

protractor in ��standard�� position, that is, a base is horizontal, even if neither arm of the angle is horizontal. Measuring and then sketching many angles with no horizontal or vertical arms,4.MD.6 perhaps ini- tially using circular 360 protractors, can help students avoid such limited conceptions.

As with length, area, and volume, children need to understand equal partitioning and unit iteration to understand angle and turn measure. Whether defined as more statically as the measure of the figure formed by the intersection of two rays or as turning, having a given angle measure involves a relationship between components of plane figures and therefore is a

4.G.2Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or ab- sence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.

Given the complexity of angles and angle measure, it is unsur-

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students integrate turns, and a general, dynamic understanding of angle measure-as-rotation, into their understandings of angles-as- objects. Computer manipulatives and tools can help children bring such a dynamic concept of angle measure to an explicit level of awareness. For example, dynamic geometry environments can pro- vide multiple linked representations, such as a screen drawing that students can ��drag�� which is connected to a numerical representa- tion of angle size. Games based on similar notions are particularly effective when students manipulate not the arms of the angle itself, but a representation of rotation (a small circular diagram with radii that, when manipulated, change the size of the target angle turned). Students with an accurate conception of angle can recognize that angle measure is

4.MD.7Recognize angle measure as additive. When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure.

when an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Students can then solve interesting and challenging addition and subtraction problems to find the measurements of unknown angles on a diagram in real world and mathematical problems. For exam- ple, they can find the measurements of angles formed a pair of inter- secting lines, as illustrated above, or given a diagram showing the measurement of one angle, find the measurement of its complement. They can use a protractor to check, not to check their reasoning, but to ensure that they develop full understanding of the mathematics and mental images for important benchmark angles (e.g., 30 , 45 , 60 , and 90 ).

Such reasoning can be challenged with many situations as il- lustrated in the margin. Similar activities can be done with drawings of shapes using right angles and half of a right angle to develop the important bench- marks of 90 and 45 . Missing measures can also be done in the turtle geometry con- text, building on the previous work. Note that unguided use of Logo��s turtle geometry does not necessary develop strong angle

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Grade 5

4.MD.1Know relative sizes of measurement units within one sys- tem of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. 5.MD.1Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems.

Convert like measurement units within a given measurement sys- tem In Grade 5, students extend their abilities from Grade 4 to express measurements in larger or smaller units within a measure- ment system.4.MD.1, 5.MD.1 This is an excellent opportunity to rein- force notions of place value for whole numbers and decimals, and connection between fractions and decimals (e.g., 2 1

2 meters can be

expressed as 2.5 meters or 250 centimeters). For example, building on the table from Grade 4, Grade 5 students might complete a table of equivalent measurements in feet and inches.

Feet Inches 0 0 1 2 3 In Grade 6, this table can be discussed in terms of ratios and proportional relationships (see the Ratio and Proportion Progression). In Grade 5, however, the main focus is on arriving at the measurements that generate the table.

Grade 5 students also learn and use such conversions in solving multi-step, real world problems (see example in the margin).

Understand concepts of volume and relate volume to multiplication and to addition The major emphasis for measurement in Grade 5 is volume. Volume not only introduces a third dimension and thus a significant challenge to students�� spatial structuring, but also com- plexity in the nature of the materials measured. That is, solid units are ��packed,�� such as cubes in a three-dimensional array, whereas a liquid ��fills�� three-dimensional space, taking the shape of the con- tainer. As noted earlier (see Overview, also Grades 1 and 3), the unit structure for liquid measurement may be psychologically one- dimensional for some students. ��Packing�� volume is more difficult than iterating a unit to measure length and measuring area by tiling. Students learn about a unit of volume, such as a cube with a side length of 1 unit, called a unit cube.5.MD.3 They pack cubes (without gaps) into right rectangular

5.MD.3Recognize volume as an attribute of solid figures and un- derstand concepts of volume measurement.

prisms and count the cubes to determine the volume or build right rectangular prisms from cubes and see the layers as they build.5.MD.4

5.MD.4Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units.

They can use the results to compare the volume of right rectangu- lar prisms that have different dimensions. Such experiences enable students to extend their spatial structuring from two to three di- mensions (see the Geometry Progression). That is, they learn to both mentally decompose and recompose a right rectangular prism built from cubes into layers, each of which is composed of rows and columns. That is, given the prism, they have to be able to de- compose it, understanding that it can be partitioned into layers, and each layer partitioned into rows, and each row into cubes. They also have to be able to compose such as structure, multiplicatively, back into higher units. That is, they eventually learn to conceptualize a layer as a unit that itself is composed of units of units—rows, each row composed of individual cubes—and they iterate that structure. Thus, they might predict the number of cubes that will be needed to fill a box given the net of the box.

Another complexity of volume is the connection between ��pack- ing�� and ��filling.�� Often, for example, students will respond that a box can be filled with 24 centimeter cubes, or build a structure of 24 cubes, and still think of the 24 as individual, often discrete, not

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5.MD.5aFind the volume of a right rectangular prism with whole- number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication.

that the height of the prism tells how many layers would fit in the prism. That is, they understand that volume is a derived attribute that, once a length unit is specified, can be computed as the product of three length measurements or as the product of one area and one length measurement. Then, students can learn the formulas

5.MD.5bApply the formulas

tencies to find the volumes of right rectangular prisms with edges whose lengths are whole numbers and solve real-world and math- ematical problems involving such prisms. Students also recognize that volume is additive (see Overview) and they find the total volume of solid figures composed of two right rectangular prisms.5.MD.5c For example, students might design

5.MD.5cRecognize volume as additive. Find volumes of solid fig- ures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems.

a science station for the ocean floor that is composed of several rooms that are right rectangular prisms and that meet a set criterion specifying the total volume of the station. They draw their station (e.g., using an isometric grid, MP7) and justify how their design meets the criterion (MP1).

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Where the Geometric Measurement Progression is heading

Connection to Geometry In Grade 6, students build on their un- derstanding of length, area, and volume measurement, learning to how to compute areas of right triangles and other special figures and volumes of right rectangular prisms that do not have measurements given in whole numbers. To do this, they use dissection arguments. These rely on the understanding that area and volume measures are additive, together with decomposition of plane and solid shapes (see the K–5 Geometry Progression) into shapes whose measurements students already know how to compute (MP1, MP7). In Grade 7, they use their understanding of length and area in learning and using formulas for the circumference and area of circles. In Grade 8, they use their understanding of volume in learning and using formulas for the volumes of cones, cylinders, and spheres. In high school, students learn formulas for volumes of pyramids and revisit the formulas from Grades 7 and 8, explaining them with dissection arguments, Cavalieri��s principle, and informal limit arguments. Connection to the Number System In Grade 6, understanding of length-units and spatial structuring comes into play as students learn to plot points in the coordinate plane. Connection to Ratio and Proportion Students use their knowledge of measurement and units of measurement in Grades 6–8, coming to see conversions between two units of measurement as describing proportional relationships.

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