First graders use non-standard objects to measure lengths to help them focus on the attribute being measured leading to the understanding of the need for a standard unit. Using multiple copies of an object, they learned to lay physical units such as centimeter or inch manipulatives end-to-end and count them to measure a length. Numerous measurement experiences should have lead students to recognize the importance of careful measuring so that there are not any gaps or overlaps in order to get an accurate measurement. Additionally, students will have previous experiences in Grade 1 using direct comparison of length, comparing the amount of length in two objects without measurement. Having developed the understanding that the results of measurement and direct comparison have the same results supported the application of measurement strategies.
Second graders build upon their non-standard measurement experiences in first grade by measuring in standard units for the first time. Using both customary
(inches and feet) and metric (centimeters and meters) units, second graders will select an attribute to be measured (e.g., length of classroom), choose an appropriate unit of measurement (e.g., yardstick), and determine the number of units (e.g., yards). As teachers provide rich tasks that ask students to perform real measurements, these foundational understandings of measurement are developed:

Understand that larger units (e.g., yard) can be subdivided into equivalent units (e.g., inches) (partition).

Understand that the same object or many objects of the same size such as paper clips can be repeatedly used to determine the length of an object
(iteration).

Understand the relationship between the size of a unit and the number of units needed (compensatory principal). Thus, the smaller the unit, the more units it will take to measure the selected attribute.
Providing students with opportunities to create and use a variety of rulers, they can connect their understanding of non-standard units from first grade to standard units in second grade. Building upon their experiences with open number lines, second grade students create number lines with evenly spaced points corresponding to the numbers to solve addition and subtraction problems to 100. They recognize the similarities between a number line and a ruler.
When second grade students measure an object using two units of different length by using the same tool and unit to measure both objects, this experience helps them realize that the unit used is as important as the attribute being measured. This is a difficult concept for young children and will require several experiences for students to predict, measure, and discuss outcomes. Students will continue to measure two objects of choice, identify an appropriate measurement tool and unit, measure both objects, and then determine the differences in lengths. Additionally, students will estimate the lengths of objects using inches, feet, centimeters, and meters prior to measuring. Estimation helps students focus on the attribute being measured and the measuring process. As students estimate, the student has to consider the size of the unit- helping them to become more familiar with the unit size. In addition, estimation also creates a problem to be solved rather than a task to be completed. Once a student has made an estimate, the student then measures the object and reflects on the accuracy of the estimate made and considers this information for the next measurement.
Major Standards Supporting Standards
Adapted from UbD Framework
Additional Standards Page 1

Furthermore, the students will use measurement data as they move through the statistical process of posing a question, collecting data, analyzing data, creating representations, and interpreting the results. In second grade students represent the length of several objects by making a line plot. Students should round their lengths to the nearest whole unit.
In Grade 3, students will develop an understanding of the concept of perimeter through various experiences, such as walking the perimeter of a room or using a geoboard. They will learn the perimeter is the boundary of a two-dimensional shape. They will explore polygons, determining the length of the perimeter is the sum of the lengths of the sides. To extend skills, they will find the perimeter of objects; use addition to find perimeters; and recognize the patterns that exist when finding the sum of the lengths and widths of rectangles. Students will also strategically use tools, such as geoboards, tiles, graph paper or technology to find all the possible rectangles that have a given perimeter (e.g., find the rectangles with a perimeter of 12 cm). They record all the possibilities using dot or graph paper, compile the possibilities into an organized list or a table, and determine whether they have all the possible rectangles. Following these experiences, students will reason about connections between their representations, side lengths, and the perimeter of the rectangles. In addition, given a perimeter and a length or width, students will use objects or pictures to find missing length or width. The use of fractional units of measurement will be included as well .
From the K-5 Geometric Measurement progression document, pp. 12, 13, 14, 15
Measure and estimate lengths in standard units second graders learn to measure length with a variety of tools, such as rulers, meter sticks, and measuring tapes.
2.MD.1
Although this appears to some adults to be relatively simple, there are many conceptual and procedural 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 understand 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.
Students also can learn accurate procedures and concepts by drawing simple unit rulers. Using copies of a single length-unit such as inch-long manipulatives, they mark off length-units on strips of paper, explicitly connecting measurement with the ruler to measurement by iterating physical units. Thus, students’ first rulers 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,
Major Standards Supporting Standards
Adapted from UbD Framework
Additional Standards Page 2

correct alignment (with a ruler), and the zero-point concept (the idea that the zero of the ruler indicates one endpoint of a length). These are reinforced as children compare the results of measuring to compare to objects with the results of directly comparing these objects.
Second graders learn the concept of the inverse relationship between the size of the unit of length and the number of units required to cover a specific length or distance.
2.MD.2
Thus, second grade students can learn that the larger the unit, the fewer number of units in a given measurement. 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.
Second graders also learn to combine and compare lengths using arithmetic operations. That is, they can add two lengths to obtain the length of the whole and subtract one length from another to find out the difference in lengths.
2.MD.4
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. These understandings are essential in supporting work with number line diagrams .2.MD.6
Students think of a number line diagram as a measurement model and use strategies relating to distance, proximity of numbers, and reference points. After experience with measuring, second graders learn to estimate lengths.
2.MD.3
Real-world applications of length often involve estimation. Skilled estimators move fluently back and forth between written or verbal length measurements and representations of their corresponding magnitudes on a mental ruler .
Length measurement should also be used in other domains of mathematics, as well as in other subjects, such as science, and connections should be made where possible. Teachers can ask students to discuss relationships they see between rulers and line plot scales. Students could also graph the results of many students measuring the same object as precisely as possible and discuss what the “real” measurement of the object might be.
Measure and estimate lengths in standard units.
2.MD.1 Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.
2.MD.2 Measure 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.
2.MD.3 Estimate lengths using units of inches, feet, centimeters, and meters.
2.MD.4 Measure to determine how much longer one object is than another, expressing length difference in terms of standard length unit.
Relate addition and subtraction to length
2.MD.6 Represent whole numbers as lengths from 0 on number line diagram with equally spaced points corresponding to the number 0, 1, 2,… and represent whole-number sums & differences within 100 on a number line diagram.
Represent and interpret data.
2.MD.9 Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.
Major Standards Supporting Standards
Adapted from UbD Framework
Additional Standards Page 3

Students will apply measurement concepts and skills to estimate and measure length appropriately and to represent that measurement in whole numbers.
Using those whole-number measurements, students will be able to represent addition and subtraction on a number line diagram.
Ex: Students will also generate measurement data by measuring lengths to create a line plot.

There is a relationship between estimation and measurement.
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A length unit is used to find a measurement.
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A specific process is used to measure attributes of unit length.
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A number line is used to represent measurement attributes such as, distance and quantity.
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When should you estimate and when do you need an exact answer?

What properties or attributes can be measured? (perimeter, length, width, height)

How do we measure (unit, tool, and process)?

How can accurate measurements help us to solve problems and make sense of our world?
Major Standards Supporting Standards
Adapted from UbD Framework
Additional Standards Page 4

1.
2.
3.
Make sense of problems and persevere in solving them . Students make sense of measurement quantities and complete a measurement project to determine length using specific units of measurement.
Reason abstractly and quantitatively . Students reason about measurement quantities and relationships while solving tasks. The process of reasoning is applied when determining length and measuring with standard units.
Construct viable arguments and critique the reasoning of others . After working on a measurement task, students will discuss and critique others’ reasoning and strategies.
*4. Model with mathematics . Students will use concrete manipulatives to measure and represent distance and quantity. Real-life addition and subtraction situations will be modeled on a number line.
*5. Use appropriate tools strategically . Students will know when to utilize measurement tools appropriately. These tools may include non-standard objects, rulers, and number lines.
*6. Attend to precision.
Students will use the measurement process precisely. For example when using non-standard objects, students will understand units must be of equal size and maintain appropriate length-until iteration. However when using a ruler, students will show correct alignment of the zero-point.
*7. Look for and make use of structure . Students will reconstruct the patterns within a measurement system by using the structure of the number line.
8. Look for and express regularity in repeated reasoning . Students will discover strategic methods for estimating length and accurately check for the reasonableness of their estimations.
Students should already be able to…

Measure using non-standard units.
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Express lengths as whole numbers within 20.
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Measure with non-standard units accurately with no gaps or overlaps.
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Order objects by length.
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Compare objects indirectly by length using a third object.
Some students may be ready to…
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Learn to subdivide length units into fractional units.

Express length as whole numbers and fractional units.

Utilize measurement to find perimeter and area.

Find unknown side lengths in “missing measurement” problems.
Major Standards Supporting Standards
Adapted from UbD Framework
Additional Standards Page 5

:

The standard tools for linear measurement.

The location of the beginning point of the appropriate standard measuring tool.

Length-units.
:
 Measure the length of an object by selecting and using appropriate

 standard tools. (2.MD.1)
Measure length of an object twice, using units of different lengths for the two measurements. (2.MD.2)
Describe how two measurements using different units relates to the size of the unit chosen. (2.MD.2)
 Estimate lengths using units of inches, feet, centimeters and meters.
(2.MD.3)
 Check for reasonableness of estimates. (2.MD.3)
 Compare objects visually, side by side, and measure the difference.

(2.MD.4)
Express the difference between lengths in terms of a standard length unit.


(2.MD.4)
Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the number 0, 1, 2. (2.MD.6)
Represent whole-number sums and differences within 100 on a number line diagram. (2.MD.6)

Generate measurement data by measuring lengths of several objects to the nearest whole unit or by making repeated measurements of the same
 object. (2.MD.9)
Show measurement data by making a line plot, where the horizontal scale is marked off in whole-number units. (2.MD.9)
English Language Learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics.
English Language Learners will benefit from:

An awareness of different measurement vocabulary.

The use of visual tools such as rulers, number diagrams or number lines, and meter sticks.

An understanding of the universality of measurement.
Major Standards Supporting Standards
Adapted from UbD Framework
Additional Standards Page 6

measure estimate ruler meter stick measuring tapes number diagram number line inch foot centimeter meter customary system metric system unit line plot linear sum difference equal
shorter longer taller wider zero compare
Major Standards Supporting Standards
Adapted from UbD Framework
Additional Standards Page 7


Assessment #1 Measuring Objects

Assessment #2 Measuring Objects
Twice

Segment #7 Measuring Length and Using Appropriate Tools

Assessment #3 Estimating Length and Measuring Objects

Assessment #4 Linear
Measurement using a Number
Line and Line Plot to Compare (2.MD.1 & 2.MD.4)

Performance Based Task (2.MD.2)
1.
Understanding Measurement Relationships Between Units (2.MD.1, 2.MD.2) (Model Lesson)
2.
Estimating Measurements (2.MD.3)
3.
Comparing Measurements and Using Number Lines to Represent Sums and Differences (2.MD.4, 2.MD.6)
4.
Using a Line Plot (2.MD.9)

Assessment #4 Linear
Measurement using a Number
Line and Line Plot

Assessment Student Self-
Assessment
Major Standards Supporting Standards
Adapted from UbD Framework
Additional Standards Page 8