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 TITLE : Unit 09: Linear Measurement SUGGESTED DURATION : 18 days

#### Unit Overview

Introduction
This unit bundles student expectations that address determining the length of an object, describing the inverse relationship between the size of the unit and the number of units needed, representing whole number distances from zero on a number line, and solving problems involving length. According to the Texas Education Agency, mathematical process standards including application, a problem-solving model, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this Unit
In Grade 1, students explored the continuous nature of linear measure by using concrete objects to measure the length of objects. Students determined the length of an object as the number of same-size units of length that, when laid end-to-end with no gaps or overlaps, reach from one end of the object to the other. Students measured the length of an object using two different units of measure, described the lengths using numbers and unit labels, and explained how and why the measurements differed. Also, in Grade 2 Unit 01, students explored locating and naming the whole numbers that correspond to specific points or locations on number lines.

During this Unit
Students begin exploring length using concrete models of standard units (inch, foot, yard, centimeter, meter, etc.) in the customary and metric measurement systems. This exploration of measuring length leads to developing an understanding of the purpose and need for using standard units of measure in society. Students use concrete tools to measure distances and record the measure to the nearest whole unit. Through the use of concrete models of standard units, students build a strong foundation for understanding the inverse relationship between the size of a unit and the number of units needed to equal the length of an object. Students review locating whole numbers on a number line and extend their understanding to representing whole number distances from zero or any given location on the number line. The relationship between the number line and standard measuring tools is applied as students transition to determining length to the nearest whole unit using rulers, yardsticks, meter sticks, and measuring tapes. Students apply their understanding of length, including estimating lengths, to problem-solving situations. Students use benchmarks to estimate solutions (e.g., a finger joint on a thumb is approximately 1 inch, the width of tip of a finger is approximately 1 centimeter, etc.) and use actual measurements to solve problems involving adding and/or subtracting lengths, including finding the distances around the outer edges of objects.

Other considerations: Reference the Mathematics COVID-19 Gap Implementation Tool Grade 2

After this Unit
In Grade 3, students will determine the perimeter of polygons or a missing length when given perimeter and remaining side lengths in problems. They will also represent fractions as distances from zero on a number line.

In Grade 2, determining the length of an object using concrete models for standard units of length and standard linear measuring tools, describing the inverse relationship between the size of the unit and the number of units needed, representing whole numbers as distances from zero on a number line, and solving problems involving length are included in the Grade 2 Texas Response to Curriculum Focal Points (TxRCFP): Measuring length. This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning A1, B1, C1; II. Algebraic Reasoning D1, D2; V. Statistical Reasoning A1, C2; VII. Problem Solving and Reasoning A1, A2, A3, A4, A5, B1, C1, D1, D2; VIII. Communication and Representation A1, A2, A3, B1, B2, C1, C2, C3; IX. Connections A1, A2, B1, B2, B3.

Research
According to Chapin and Johnson (2000), “In order to prepare elementary students to use measurement in a variety of ways, we must provide them with many opportunities to engage in meaningful measurement tasks – tasks that promote understanding of concepts and facility with measurement units” (p.195). As teachers, we try to facilitate fluency and accuracy using measurement tools. Van de Walle (2005) explains that the reason students struggle with measurement is that they do not understand how the measuring device works. He suggests that, “If students actually make simple measuring instruments using unit models with which they are familiar, it is more likely that they will understand how an instrument measures” (p.255-56). He also suggests, “It is essential that the informal instrument be compared with the standard instrument. Without this comparison, students may not understand that these two instruments are really two means to the same end” (p.255-56).

Chapin, S & Johnson, A. (2000). Math matters: Understanding the math you teach. Sausalito, CA: Math Solutions Publications
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9
Texas Education Agency. (2013). Texas response to curriculum focal points for kindergarten through grade 8 mathematics. Retrieved from https://www.texasgateway.org/resource/txrcfp-texas-response-curriculum-focal-points-k-8-mathematics-revised-2013
Van de Walle, J., & Lovin, L. (2005). Teaching student-centered mathematics grades 3 – 5. Boston, MA: Pearson Education, Inc.

 Geometric, spatial, and measurement reasoning are foundational to visualizing, analyzing, and applying relationships within and between scale, shapes, quantities, and spatial relations in everyday life. Why is developing geometric, spatial, and measurement reasoning essential? How does geometric, spatial, and measurement reasoning affect how one sees and works in the world?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Objects have unique measurable attributes that can be defined and described in order to make sense of their relationship to other objects in the world.
• Why is it important to be able to measure length?
• In what situations might someone need to measure length?
• Attributes of objects can be measured using tools, and their measures can be described using units, in order to quantify a measurable attribute of the object.
• What tools can be used to measure length?
• How are tools used to measure length?
• Why is it important to …
• use the same-sized units to measure length?
• determine the starting and ending points of the length being measured?
• not allow gaps between units when measuring length?
• not allow overlaps of units when measuring length?
• How can composition or decomposition be used to simplify the measurement process?
• How can the length of an object be determined when the end point does not align with the last unit?
• How can a length that is not straight be measured?
• How can the length of an object be described?
• What relationships exist between
• the size of the unit and the number of units needed to equal the length of an object?
• length and distance?
• length and height?
• length and width?
• Measurement
• Measureable Attributes
• Distance and length
• Measure
• Measurement tools
• Units of measure
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

 Geometric, spatial, and measurement reasoning are foundational to visualizing, analyzing, and applying relationships within and between scale, shapes, quantities, and spatial relations in everyday life. Why is developing geometric, spatial, and measurement reasoning essential? How does geometric, spatial, and measurement reasoning affect how one sees and works in the world?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Objects have unique measurable attributes that can be defined and described in order to make sense of their relationship to other objects in the world.
• Why is it important to be able to measure length?
• In what situations might someone need to measure length?
• Attributes of objects can be measured using tools, and their measures can be described using units, in order to quantify a measurable attribute of the object.
• What tools can be used to measure length?
• How are tools used to measure length?
• How can composition or decomposition be used to simplify the measurement process?
• How can the length of an object be determined when the end point does not align with the last unit?
• How can a length that is not straight be measured?
• How can the length of an object be described?
• What relationships exist between …
• the position of a whole number on a number line and distance?
• length and distance?
• length and height?
• length and width?
• rulers, yardsticks, meter sticks, and measuring tapes?
• How are rulers, yardsticks, meter sticks, and measuring tapes alike and different?
• Measurement
• Measureable Attributes
• Distance and length
• Measure
• Measurement tools
• Units of measure
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think the numbers on a ruler are counting marks rather than numbered units that indicate space or distance between the marks.
• Some students may think when measuring linear length, the edge of the ruler is placed at the starting point instead of at the zero mark on the ruler.
• Some students may think measurement using a ruler must begin at the zero mark rather than realizing measurement can begin at any interval on a ruler, and the new number now becomes the zero to count intervals from.
• Some students may think when measuring length to the closest whole number, the last unit is either always counted or never counted rather than recognizing the whole unit the end mark is actually closest to determines whether or not to count the unit.
• Some students may think customary and metric measurements can be combined rather than understanding these are two separate measurement systems.
• Some students may think the longer the unit, the larger the count and vice versa rather than understanding the longer the unit, the fewer units needed and vice versa.

Underdeveloped Concepts:

• Some students may not understand that units must be of equal size when measuring with concrete objects.
• Some students may leave gaps between units or overlap units when measuring with concrete objects.
• Some students may struggle with selecting an appropriate unit of measure for efficiency and/or precision.
• Some students may struggle, when measuring with concrete objects, determining if the last unit is greater than or less than one-half, and whether it should be counted.

#### Unit Vocabulary

• Estimation – reasoning to determine an approximate value
• Length – the measurement attribute that describes a continuous distance from end to end
• Linear measurement – the measurement of length along a continuous line or curve
• Unit of length – the object or unit used to measure length
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}

Related Vocabulary:

 Approximate Centimeter Continuous Customary units of measure Decimeter Distance Foot Height Horizontal Inch Increments Measurement attribute Measuring tape Meter Meter stick Metric units of measure Number line Precision Ruler Scale Vertical Yard Yardstick
Unit Assessment Items System Resources Other Resources

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Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Center if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards

Texas Education Agency – Texas Response to Curriculum Focal Points for K-8 Mathematics Revised 2013

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Grade 2 Mathematics TEKS

TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE Legend:

• Knowledge and Skills Statements (TEKS) identified by TEA are in italicized, bolded, black text.
• Student Expectations (TEKS) identified by TEA are in bolded, black text.
• Portions of the Student Expectations (TEKS) that are not included in this unit but are taught in previous or future units are indicated by a strike-through.

Specificity Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
TEKS# SE# TEKS SPECIFICITY
2.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
2.1A Apply mathematics to problems arising in everyday life, society, and the workplace.

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE

Including, but not limited to:

• Mathematical problem situations within and between disciplines
• Everyday life
• Society
• Workplace

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing proficiency in the use of place value within the base-10 numeration system
• Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Measuring length
• Applying knowledge of two-dimensional shapes and three-dimensional solids, including exploration of early fraction concepts
• TxCCRS:
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world situation.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
• IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.
2.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION

Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving process and the reasonableness of the solution

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing proficiency in the use of place value within the base-10 numeration system
• Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Measuring length
• Applying knowledge of two-dimensional shapes and three-dimensional solids, including exploration of early fraction concepts
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.A. Statistical Reasoning – Design a study
• V.A.1. Formulate a statistical question, plan an investigation, and collect data.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.A.2. Formulate a plan or strategy.
• VII.A.3. Determine a solution.
• VII.A.4. Justify the solution.
• VII.A.5. Evaluate the problem-solving process.
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.2. Evaluate the problem-solving process.
2.1C Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS

Including, but not limited to:

• Appropriate selection of tool(s) and techniques to apply in order to solve problems
• Tools
• Real objects
• Manipulatives
• Paper and pencil
• Technology
• Techniques
• Mental math
• Estimation
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing proficiency in the use of place value within the base-10 numeration system
• Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Measuring length
• Applying knowledge of two-dimensional shapes and three-dimensional solids, including exploration of early fraction concepts
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
• V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
2.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE

Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• Graphs
• Language

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing proficiency in the use of place value within the base-10 numeration system
• Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Measuring length
• Applying knowledge of two-dimensional shapes and three-dimensional solids, including exploration of early fraction concepts
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
2.1E Create and use representations to organize, record, and communicate mathematical ideas.

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Representations of mathematical ideas
• Organize
• Record
• Communicate
• Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
• Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing proficiency in the use of place value within the base-10 numeration system
• Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Measuring length
• Applying knowledge of two-dimensional shapes and three-dimensional solids, including exploration of early fraction concepts
• TxCCRS:
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
2.1F Analyze mathematical relationships to connect and communicate mathematical ideas.

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing proficiency in the use of place value within the base-10 numeration system
• Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Measuring length
• Applying knowledge of two-dimensional shapes and three-dimensional solids, including exploration of early fraction concepts
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
•  VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
2.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION

Including, but not limited to:

• Mathematical ideas and arguments
• Validation of conclusions
• Displays to make work visible to others
• Diagrams, visual aids, written work, etc.
• Explanations and justifications
• Precise mathematical language in written or oral communication

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing proficiency in the use of place value within the base-10 numeration system
• Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Measuring length
• Applying knowledge of two-dimensional shapes and three-dimensional solids, including exploration of early fraction concepts
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.4. Justify the solution.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
2.9 Geometry and measurement. The student applies mathematical process standards to select and use units to describe length, area, and time. The student is expected to:
2.9A Find the length of objects using concrete models for standard units of length.

Find

THE LENGTH OF OBJECTS USING CONCRETE MODELS FOR STANDARD UNITS OF LENGTH

Including, but not limited to:

• Length – the measurement attribute that describes a continuous distance from end to end
• Unit of length – the object or unit used to measure length
• Concrete models that represent standard units of length
• Typically used customary units of length
• Inch represented by a color tile, etc.
• Foot represented by a 12-inch ruler as a single unit, etc.
• Yard represented by a yardstick as a single unit, etc.
• Typically used metric units of length
• Centimeter represented by a base-10 unit cube, etc.
• Decimeter represented by a base-10 long, orange Cuisenaire rod, etc.
• Meter represented by a meter stick as a single unit, etc.
• Length described to the nearest whole unit using a number and a unit
• Linear measurement – the measurement of length along a continuous line or curve
• Starting point and ending point defined
• Equal sized units of length placed end to end along the distance being measured
• Equal sized units of length iterated (repeated) with no gaps or overlaps
• Length measured using one-dimensional units of length (e.g., if measuring with a color tile, measure with the edge, not the area of the color tile; if measuring with a color tile, measure with the same dimension of the color tile; etc.)
• Equal sized units of length counted to the nearest whole unit
• Last unit is not counted if the end point falls less than half-way along the unit.
• Last unit is counted if the end point falls half-way, or more than half-way, along the unit.
• Unit of length selected for efficiency
• Smaller unit of length to measure shorter objects or distances
• Larger unit of length to measure longer objects or distances
• Unit of length selected for precision
• Smaller unit of length results in a more precise measurement when measuring to the whole unit
• Larger unit of length results in a less precise measurement when measuring to the whole unit

Note(s):

• Grade 1 illustrated that the length of an object is the number of same-size units of length that, when laid end-to-end with no gaps or overlaps, reach from one end of the object to the other.
• Grade 1 described a length to the nearest whole unit using a number and a unit.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Measuring length
• TxCCRS:
• I.C. Numeric Reasoning – Systems of measurement
• I.C.1. Select or use the appropriate type of method, unit, and tool for the attribute being measured.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
2.9B Describe the inverse relationship between the size of the unit and the number of units needed to equal the length of an object.

Describe

THE INVERSE RELATIONSHIP BETWEEN THE SIZE OF THE UNIT AND THE NUMBER OF UNITS NEEDED TO EQUAL THE LENGTH OF AN OBJECT

Including, but not limited to:

• Length – the measurement attribute that describes a continuous distance from end to end
• Unit of length – the object or unit used to measure length
• Concrete models that represent standard units of length
• Typlically used customary units of length
• Inch represented by a color tile, etc.
• Foot represented by a 12-inch ruler as a single unit, etc.
• Yard represented by a yardstick as a single unit, etc.
• Typically used metric units of length
• Centimeter represented by a base-10 unit cube, etc.
• Decimeter represented by a base-10 long, orange Cuisenaire rod, etc.
• Meter represented by a meter stick as a single unit, etc.
• Inverse relationship between the size of the unit and the number of units needed
• Measure the same object with different sized units of length.
• The shorter the unit of length, the more units needed
• The longer the unit of length, the fewer units needed

Note(s):

• Grade 1 measured the same object/distance with units of two different lengths and described how and why the measurements differ.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Measuring length
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.1. Compare relative magnitudes of rational and irrational numbers, and understand that numbers can be represented in different ways.
• I.C. Numeric Reasoning – Systems of measurement
• I.C.1. Select or use the appropriate type of method, unit, and tool for the attribute being measured.
2.9C Represent whole numbers as distances from any given location on a number line.

Represent

WHOLE NUMBERS AS DISTANCES FROM ANY GIVEN LOCATION ON A NUMBER LINE

Including, but not limited to:

• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
• Characteristics of a number line
• A number line begins as a line with predetermined intervals (or tick marks) with positions/numbers labeled.
• A minimum of two positions/numbers should be labeled.
• Numbers on a number line represent the distance from zero.
• The distance between the tick marks is counted rather than the tick marks themselves.
• The placement of the labeled positions/numbers on a number line determines the scale of the number line.
• Intervals between position/numbers are proportional.
• When reasoning on a number line, the position of zero may or may not be placed.
• When working with larger numbers, a number line without the constraint of distance from zero allows the ability to “zoom-in” on the relevant section of the number line.
• Number lines extend infinitely in both directions (arrows indicate the number line continues infinitely).
• Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number line.
• Points to the left of a specified point on a horizontal number line are less than points to the right.
• Points to the right of a specified point on a horizontal number line are greater than points to the left.
• Points below a specified point on a vertical number line are less than points above.
• Points above a specified point on a vertical number line are greater than points below.
• Whole numbers represented as equally spaced lengths or distances from zero on a number line
• Relationship between a whole number represented using a strip diagram to a whole number represented on a number line
• Number lines beginning with a number other than zero
• Distance from zero to first marked increment is assumed even when not visible on the number line.
• Relationship between whole numbers as distances from zero on a number line to whole unit measurements as distances from zero on a customary ruler, yardstick, or measuring tape
• Measuring a specific length using a distance other than zero on a ruler, yardstick, or measuring tape
• Distance from zero to first marked increment not counted
• Length determined by number of whole units between starting point and ending point

Note(s):

• Grade 3 will represent fractions of halves, fourths, and eighths as distances from zero on a number line.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Measuring length
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.1. Compare relative magnitudes of rational and irrational numbers, and understand that numbers can be represented in different ways.
2.9D Determine the length of an object to the nearest marked unit using rulers, yardsticks, meter sticks, or measuring tapes.

Determine

THE LENGTH OF AN OBJECT TO THE NEAREST MARKED UNIT USING RULERS, YARDSTICKS, METER STICKS, OR MEASURING TAPES

Including, but not limited to:

• Length – the measurement attribute that describes a continuous distance from end to end
• Unit of length – the object or unit used to measure length
• Linear measurement – the measurement of length along a continuous line or curve
• Standard linear measurement tools
• Typically used customary linear measurement tools
• Ruler with inches, yardstick, measuring tape
• Typically used metric linear measurement tools
• Ruler with centimeters, meter stick, measuring tape
• Standard units of length
• Typically used customary units of length
• Inches, feet, yards
• Typically used metric units of length
• Centimeters, meters
• Relationship between finding the length of objects using concrete models for standard units of length to whole unit measurements on a customary ruler, yardstick, or measuring tape
• Relationship between whole numbers as distances from zero on a number line to whole unit measurements as distances from zero on a customary ruler, yardstick, or measuring tape
• Determine length to the nearest whole unit.
• Starting point and ending point defined
• Edge of measuring tool placed along the distance being measured, aligned with the start point of the distance being measured
• Equal sized units of length counted to the nearest whole unit
• Last unit is not counted if the end point falls less than half-way along the unit.
• Last unit is counted if the end point falls half-way, or more than half-way, along the unit.
• Measuring a specific length using a starting point other than zero on a ruler, yardstick, or measuring tape
• Distance from zero to first marked increment not counted
• Length determined by number of whole units between starting point and ending point
• Selection of appropriate tool and unit of length in real-world situations

Note(s):

• Grade 1 used non-standard measuring tools to measure the length of objects to reinforce the continuous nature of linear measurement.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Measuring length
• TxCCRS:
• I.C. Numeric Reasoning – Systems of measurement
• I.C.1. Select or use the appropriate type of method, unit, and tool for the attribute being measured.
2.9E Determine a solution to a problem involving length, including estimating lengths.

Determine

A SOLUTION TO A PROBLEM INVOLVING LENGTH, INCLUDING ESTIMATING LENGTHS

Including, but not limited to:

• Length – the measurement attribute that describes a continuous distance from end to end
• Mathematical and real-world problem situations
• Recognition of attributes of length embedded in mathematical and real-world problem situations (e.g., distance traveled from one place to another, length of an object, distance around the outer edges of an object, etc.)
• One-step or multi-step problems
• Measurement of one or more distances/lengths
• Multiple operations
• Addition and/or subtraction of whole unit measurements
• Solutions recorded to the nearest whole unit with a number and a unit label
• Estimation – reasoning to determine an approximate value
• Estimation prior to solving problem
• Estimation compared to actual measurement
• Benchmarks for units of length
• Finger joint (thumb works best) = approximately 1 inch
• Tip of your finger = approximately 1 centimeter
• Span of your palm = approximately 1 decimeter
• Elbow to wrist = approximately 1 foot
• Nose to fingertip of extended arm = approximately 1 yard
• Nose to fingertip of extended arm with head turned away = approximately 1 meter
• Language related to estimation
• About, a little less than, a little more than, almost, nearly, approximately, etc.

Note(s): 