# 1.3.2A Length

1
Subject:
Math
Strand:
Geometry & Measurement
Standard 1.3.2

Use basic concepts of measurement in real-world and mathematical situations involving length, time and money.

Benchmark: 1.3.2.2 Time

Tell time to the hour and half-hour.

Benchmark: 1.3.2.1 Length

Measure the length of an object in terms of multiple copies of another object.

For example: Measure a table by placing paper clips end-to-end and counting.

## Overview

Big Ideas and Essential Understandings

Essential Understandings/Ideas

First graders move beyond direct comparison as they develop an understanding of linear measurement. They measure length by laying multiple copies of a unit end to end and counting the units. For example, the pencil is eight blocks long. First graders come to understand that using a larger unit when measuring means you will use fewer units and using a smaller unit when measuring means you will use more units. When objects are being compared they know that the units used to measure each object must be the same size.

When First Graders use calendars or sequence events in stories, they are using measures of time in a real context. First graders learn to tell time to the hour and half-hour using analog and digital clocks. They relate these times to events during their day. For example, we eat lunch at 12:30pm.

First graders identify and know the value of dimes, nickels and pennies. They are able to count groups of dimes, nickels and pennies up a dollar.

All Standard Benchmarks - with codes

1.3.2.1 Measure the length of an object in terms of multiple copies of another object.

1.3.2.2 Tell time to the hour and half-hour.

1.3.2.3 Identify pennies, nickels and dimes; find the value of a group of these coins, up to one dollar.

Benchmark Cluster

1.3.2.1 Measure the length of an object in terms of multiple copies of another object.

What students should know and be able to do [at a mastery level] related to this benchmark:

• Measure the length of an object using non-standard units of measurement.
• understand that the size of the unit of measure is related to the number of units needed to measure a given object
• understand units need to be the same size when measuring
• understand how units are placed when measuring the length of an object.

Work from previous grades that supports this new learning includes:

• Use words such as shorter, longer/taller, and about the same to compare the length of objects. Use this knowledge to order the objects.
• Use words such as larger, smaller, and about the same to compare the size of objects. Use this knowledge to order the objects.
• Use words such as heavier, lighter, and about the same to compare weight of objects. Use this knowledge to order the objects.
Correlations

NCTM Standards

Understand measurable attributes of objects and the units, systems, and processes of measurement

PreK-2 Expectations

• recognize the attributes of length, volume, weight, area, and time.
• compare and order objects according to these attributes.
• understand how to measure using nonstandard and standard units.
• select an appropriate unit and tool for the attribute being measured.

Apply appropriate techniques, tools, and formulas to determine measurements

PreK-2 Expectations

• measure with multiple copies of units of the same size, such as paper clips laid end to end.
• use repetition of a single unit to measure something larger than the unit, for instance, measuring the length of a room with a single meterstick.

Common Core State Standards:

Measure lengths indirectly and by iterating length units.

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

1.MD.2. Express 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. Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.

Tell and write time.

1.MD.3. Tell and write time in hours and half-hours using analog and digital clocks.

## Misconceptions

Student Misconceptions

Students may think . . .

• it is OK to overlap units or leave gaps between units when measuring.
• they can start and end units anywhere when measuring the length of an object.
• units used to measure can be of mixed sizes.

## Resources

Instructional Notes

Teacher Notes

• Students may need support in further development of previously studied concepts and skills.
• Providing opportunities to measure important objects in familiar settings strengthens counting and number sense.
• "Most researchers agree there are four components of measuring:
• conservation (objects maintain their same size and shape when measured),
• transitivity (two objects can be compared in terms of a measurable quality, using another object),
• units (the type of units used to measure an object depends on the attribute being measured), and
• unit iteration (the units must be repeated, or iterated, in order to to determine the measure of an object." (Chapin, Johnson, 2006.)

To measure an attribute of an object with understanding, students should complete three steps:

1. Decide on the attribute (length) to be measured.

2. Select an appropriate unit for measuring that attribute (length).

3. Count the number of units needed to accurately represent the attribute (length) being measured.

• Measurement involves a comparison of an item that is being measured with a unit that has the same attribute (length, weight, etc). To measure anything meaningfully, the attribute being measured must be understood.
• Encourage the use of approximate language. The pencil is about 7 paper clips long. The chair is a little more than 10 straws high.  The use of approximate language is helpful because most measurements do not come out evenly.
• It is important to measure the same object with different sized non-standard units. Students will come to understand that the length of the object being measured does not change even though it will take more of some units and less of others to describe the length of the object.

Questioning

• Good questions, and good listening, will help children make sense of the mathematics, build self-confidence and encourage mathematical thinking and communication. A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."

Getting Started

What do you need to find out?

What do you know now? How can you get the information? Where can you begin?

What terms do you understand/not understand?

What similar problems have you solved that would help?

While Working

How can you organize the information?

Can you make a drawing (model) to explain your thinking? What are other possibilities?

What would happen if...?

Can you describe an approach (strategy) you can use to solve this?

What do you need to do next?

Do you see any patterns or relationships that will help you solve this?

How does this relate to...?

Why did you...?

What assumptions are you making?

What did you try that did not work? Has the question been answered?

Can the explanation be made clearer?

Responding (helps clarify and extend their thinking)

Tell me more.

Can you explain it in a different way?

Is there another possibility or strategy that would work?

Is there a more efficient strategy?

Help me understand this part ...

(Adapted from They're Counting on Us, California Mathematics Council, 1995)

Instructional Resources
• Making a "Measuring Stick"

Have the children take a strip of adding machine paper and tape to it end to end anything they would like to measure (examples: paper clips, popsicle sticks, q-tips, chips, etc.). Once they have the tape, have them label the number of items they have.  Now they can measure things with it, similar to a ruler.  Have them exchange tapes to see how the different things they used will make a difference in their measuring answer.

These are some examples of three children starting their sticks

Other Resources

This unit engages students in activities that focus on measurement and geometry. Students connect what they see and do each day with practical uses of mathematics. Maps are tools that incorporate mathematical concepts and show spatial relationships, principles of location, and navigation.

The sequence of lessons in this unit plan builds skills for measuring with non-standard units and describing, naming, interpreting, and representing relative positions in space. Students investigate various areas and objects they see each day to apply ideas of navigating in space and understanding the relationships among these various elements.

In this lesson, students participate in activities to develop concepts of measurement and statistics. Students are asked to measure distances using non-standard units and to record their measurements in a bar graph. Then they are asked to make comparisons using the bar graph.

This lesson focuses students' attention on the attributes of length and develops their knowledge of and skill in using nonstandard units of measurement. It provides practice with and remediation of the measurable attributes of length.

Dacey, L., Cavanagh, M., Findell, C., Greenes, C., Scheffield, L., & Small, M. (2003). Navigating through measurement in prekindergarten-grade 2. Reston, VA: National Council of Teachers of Mathematics.

Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction.  New York, NY: Teachers College Press.

Van de Walle, J., Karp, K., Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.). Boston, MA:  Allyn & Bacon.

Van de Walle, J., & Lovin, L.  (2006). Teaching student-centered mathematics grades k-3. Boston, MA: Pearson Education.

New Vocabulary

New Vocabulary

"Vocabulary literally is the key tool for thinking."

Ruby Payne

nonstandard unit:  units other than customary units used for measurement, examples: hand, straws, paper clips, foot, cubes, etc.

standard unit:  a standard amount or quantity, examples: inch, foot, yard, etc.

measure:  use of standard units to find out size or quantity in regard to length, height, area, mass, weight, volume, capacity, temperature and time

Mathematics vocabulary words describe mathematical relationships and concepts and cannot be understood by simply practicing definitions.  Students need to have experiences communicating ideas using these words to explain, support, and justify their thinking.

Learning vocabulary in the mathematics classroom is contingent upon the following:

Integration:   Connecting new vocabulary to prior knowledge and previously learned vocabulary.  The brain seeks connections and ways to make meaning which occurs when accessing prior knowledge.

Repetition:    Using the word or concept many times during the learning process and connecting the word or concept with its meaning.  The role of the teacher is to provide experiences that will guarantee connections are made between mathematical concepts, relationships, and corresponding vocabulary words.

Meaningful    Multiple and varied opportunities to use the words in context.  These

Use:              opportunities occur when students explain their thinking, ask clarifying questions, write about mathematics, and think aloud when solving problems.  Teachers should be constantly probing student thinking in order to determine if students are connecting mathematics concepts and relationships with appropriate mathematics vocabulary.

Strategies for vocabulary development

Students do not learn vocabulary words by memorizing and practicing definitions. The following strategies keep vocabulary visible and accessible during instruction.

Mathematics Word Bank:  Each unit of study should have word banks visible during instruction.  Words and corresponding definitions are added to the word bank as the need arises.  Students refer to word banks when communicating mathematical ideas which leads to greater understanding and application of words in context.

Labeled pictures and charts:  Diagrams that are labeled provide opportunities for students to anchor their thinking as they develop conceptual understanding and increase opportunities for student learning.

Frayer Model: The Frayer Model connects words, definitions, examples and non-examples.

Example/Non-example Charts: This graphic organizer allows students to reason about mathematical relationships as they develop conceptual understanding of mathematics vocabulary words.  Teachers should use these during the instructional process to engage student in thinking about the meaning of words.

Vocabulary Strips:  Vocabulary strips give students a way to organize critical information about mathematics vocabulary words.

 word definition illustration

Encouraging students to verbalize thinking by drawing, talking, and writing increases opportunities to use the mathematics vocabulary words in context.

Murray, M. (2004). Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.

Sammons, L. (2011).  Building mathematical comprehension: Using literacy strategies to make meaning.  Huntington Beach, CA: Shell Education.

Professional Learning Communities

Reflection - Critical Questions regarding the teaching and learning of these benchmarks

What are the key ideas related to measuring length at the first grade level?

How do student misconceptions interfere with mastery of these ideas?

What experiences do students need in order to measure length successfully using non-standard units?

When checking for student understanding of measuring length, what should teachers

• listen for in student conversations?
• look for in student work?

Examine student work related to a task involving measuring length using non-standard units.  What evidence do you need to say a student is proficient?   Using three pieces of student work, determine what student understanding is observed through the work.

How can teachers assess student learning related to these benchmarks?

How are these benchmarks related to other benchmarks at the first grade level?

Materials

Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K.. (2010). Math misconceptions prek-grade 5: from misunderstanding to deep understanding.  Portsmouth, NH: Heinemann.

Chapin, S. and Johnson, A. (2006).  Math matters, understanding the math you teach, Grades K-8, 2nd Edition. Sausalito, CA: Math Solutions Press.

Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). Classroom discussions:  using math talk to help students learn (Grades K-6). Sausalito, CA: Math Solutions.

Fuson, K., Clements, D., & Beckmann, S.. (2009). Focus in Grade1: Teaching with Curriculum Focal Points. Reston, VA: National Council of Teachers of Mathematics.

Hyde, Arthur. (2006) Comprehending math adapting reading strategies to teach mathematics, K-6. Portsmouth, NH: Heinemann.

Lester, F. (2010). Teaching and learning mathematics: transforming research for elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.

Sammons, L., (2011). Building mathematical comprehension-using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.

References

Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K.. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding.  Portsmouth, NH: Heinemann.

Bender, W. (2009). Differentiating math instruction-Strategies that work for k-8 classrooms!. Thousand Oaks, CA.: Corwin Press.

Bresser, R., Melanese, K., & Sphar, C. (2008).  Supporting English language learners in math class, grades k-2. Sausalito, CA: Math Solutions Publications.

Burns, M. (Ed.). (1998).  Leading the way: Principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.

Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.

Chapin, S. and Johnson, A. (2006).  Math matters, understanding the math you teach, Grades K-8, 2nd Edition. Sausalito, CA: Math Solutions Press.

Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). Classroom discussions: Using math talk to help students learn (Grades K-6). Sausalito, CA: Math Solutions.

Dacey, L., Cavanagh, M., Findell, C., Greenes, C., Scheffield, L., & Small, M. (2003). Navigating through measurement in prekindergarten-grade 2. Reston, VA: National Council of Teachers of Mathematics.

Dacey, L. & Salemi, R. (2007). Math for all: Differentiating instruction k-2. Sausalito, CA: Math Solutions.

Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC.: National Academies Press.

Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.

Fuson, K., Clements, D., & Beckmann, S. (2009).  Focus in grade1: Teaching with curriculum focal points .Reston, VA: National Council of Teachers of Mathematics.

K-6. Portsmouth, NH: Heinemann.

Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.

Leinwand, S., (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH:  Heinemann.

Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.

Lester, F. (2010). Teaching and learning mathematics: Transforming research for school administrators. Reston, VA:  National Council of Teachers of Mathematics.

Murray, M. & Jorgensen, J. (2007).  The differentiated math classroom-a guide for teachers k-8. Portsmouth, NH: Heinemann.

Murray, M. (2004). Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.

National Council of Teachers of Mathematics. (2000).  Principles and Standards for school mathematics.  Reston, VA:  NCTM.

Sammons, L., (2011). Building mathematical comprehension-Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.

Schielack, J., Charles, R., Clements, D., Duckett, P., Fennell, F., Lewandowski, S., ...., & Zbiek, R. M., (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: NCTM.

Seeley, C. (2009). Faster isn't smarter-Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.

Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction.  New York, NY: Teachers College Press.

Van de Walle, J., Karp, K., & Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.) Boston, MA: Allyn & Bacon.

Van de Walle, J., & Lovin, L. (2006). Teaching student-centered mathematics grades k-3. Boston, MA: Pearson Education.

West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.

## Assessment

Ask students to measure to find the following:

How many snap cubes is the longest side of your desk?

How many paper clips is the shortest side of your desk?

Find something that is shorter than five paper clips.

Find something that is between six and ten snap cubes long.

Solution:          Correctly measures and finds objects that fit the descriptions.

Benchmark:     1.3.2.1

## Differentiation

Struggling Learners

Struggling students need to use larger units when first measuring length. Larger units means fewer units which, in turn, means smaller numbers when they compare the length of two objects.

Students may struggle when aligning units. They may overlap units or leave gaps when placing units to determine the length of an object. Teacher rmodelling will help students understand the placement of units when measuring.

Students will need to measure the length of an object using several different size units in order to understand that the size of the unit is directly related to the number of units needed when measuring length.

(Adapted from The Access Center: Improving Access for All K-8 Students)

The Concrete-Representational-Abstract Instructional Approach (CRA) is a research-based instructional strategy that has proven effective in enhancing the mathematics performance of students who struggle with mathematics.

The CRA approach is based on three stages during the learning process:

Concrete     -     Representational     -     Abstract

The Concrete Stage is the doing stage. The concrete stage is the most critical in terms of developing conceptual understanding of mathematical skills and concepts.  At this stage, teachers use manipulatives to model mathematical concepts. The physical act of touching and moving manipulatives enables students to experience the mathematical concept at a concrete level.  Research shows that students who use concrete materials develop more precise and comprehensive mental representations, understand and apply mathematical concepts, and are more motivated and on-task.   Manipulatives must be selected based upon connections to the mathematical concept and the students' developmental level.

The Representational Stage is the drawing stage.  Mathematical concepts are represented using pictures or drawings of the manipulatives previously used at the Concrete Stage. Students move to this level after they have successfully used concrete materials to demonstrate conceptual understanding and solve problems.  They are moving from a concrete level of understanding toward an abstract level of understanding when drawing or using pictures to represent their thinking.  Students continue exploring the mathematical concept at this level while teachers are asking questions to elicit student thinking and understanding.

The Abstract Stage is the symbolic stage.  Teachers model mathematical concepts using numbers and mathematical symbols.  Operation symbols are used to represent addition, subtraction, multiplication and division.  Some students may not make a clean transfer to this level. They will work with some symbols and some pictures as they build abstract understanding.  Moving to the abstract level too quickly causes many student errors.   Practice at the abstract level will not lead to increased understanding unless students have a foundation based upon concrete and pictorial representations.

Bender, W. (2009). Differentiating math instruction-Strategies that work for k-8 classrooms!. Thousand Oaks, CA: Corwin Press.

Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades k-2. Sausalito, CA: Math Solutions.

Murray, M. & Jorgensen, J. (2007). The differentiated math classroom-A guide for teachers k-8. Portsmouth, NH: Heinemann.

Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction.  New York, NY: Teachers College Press.

Van de Walle, J., Karp, K., & Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.). Boston, MA: Allyn & Bacon.

Van de Walle, J. & Lovin, L. (2006). Teaching student-centered mathematics grades k-3. Boston, MA: Pearson Education.

English Language Learners

Measuring the length of several objects with various units provides an opportunity to incorporate language with the action of measuring. Terms such as unit and length need to be highlighted throughout instruction.

• Word banks need to be part of the student learning environment in every mathematics unit of study.
• Use vocabulary graphic organizers such as the Frayer model (see below) to emphasize vocabulary words count, first, second, third, etc.

• Sentence Frames

Math sentence frames provide support that English Language Learners need in order to fully participate in math discussions.  Sentence frames provide appropriate sentence structure models, increase the likelihood of responses using content vocabulary, help students to conceptualize words and build confidence in English Language Learners.

Sample sentence frames related to these benchmarks:

 The _____________ is _____________  paper clips long.
 I measured _____________________.  It is __________________________ long.
 The _____________ is shorter than the _____________ because __________________.
 The _______________  is longer than the  _______________ because _____________.
• When assessing the math skills of an ELL student it is important to determine if the student has difficulty with the math concept or with the language used to describe the concept and conceptual understanding.

Bresser, R., Melanese, K., & Sphar, C. (2008).  Supporting English language learners in math class, grades k-2. Sausalito, CA: Math Solutions Publications.

Extending the Learning

Students should measure the height of objects in the room: the reading table, the seat of a chair, etc. How do they measure?  What units do they use?  What happens if the unit used is not stackable; e.g., popsicle sticks, etc.

Bender, W. (2009). Differentiating math instruction-Strategies that work for k-8 classrooms!. Thousand Oaks, CA.: Corwin Press.

Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades k-2. Sausalito, CA: Math Solutions.

Murray, M., & Jorgensen, J. (2007). The differentiated math classroom-A guide for teachers k-8. Portsmouth, NH: Heinemann.

Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction.  New York, NY: Teachers College Press.

Classroom Observation
 Students are ... Teachers are ... correctly measuring length using a variety of units-paper clips, straws, popsicle sticks, snap cubes, etc. observing the students' use of non standard units of measure and asking questions including What are you measuring? What unit are you using to measure? Would you need more _____ or ____ to measure the length of _________. measuring the length of the same object. using different units. challenging students by matching specific units to specific objects during measurement tasks.

For Mathematics Coaches

Chapin, S. and Johnson, A. (2006).  Math matters: Understanding the math you teach: Grades k-8, 2nd edition.  Sausalito, CA: Math Solutions.

Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.

Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.

Sammons, L., (2011).  Building mathematical comprehension: Using literacy strategies to make meaning.  Huntington Beach, CA: Shell Education.

West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.

Burns, M. (Ed). (1998).  Leading the way: Principals and superintendents look at math instruction. Sausalito, CA:  Math Solutions.

Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.

Leinwand, S. (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH:  Heinemann.

Lester, F. (2010). Teaching and learning mathematics: Transforming research for school administrators. Reston, VA:  National Council of Teachers of Mathematics.

Seeley, C. (2009). Faster isn't smarter: Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions

Parents

Mathematics handbooks to be used as home references:

Cavanagh, M. (2004). Math to Know: A mathematics handbook. Wilmington, MA:  Great Source Education Group, Inc.

Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA:  Great Source Education Group, Inc.

Provides activities for children in preschool through grade 5

What should I look for in the mathematics program in my child's school? A Guide for Parents developed by SciMathMN

In helping children learn, one goal is to assist children in becoming critical and independent thinkers. You can help by asking questions that guide, without telling them what to do.

Good questions, and good listening, will help children make sense of the mathematics, build self-confidence and encourage mathematical thinking and communication. A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."

Getting Started

What do you need to find out?

What do you know now? How can you get the information? Where can you begin?

What terms do you understand/not understand?

What similar problems have you solved that would help?

While Working

How can you organize the information?

Can you make a drawing (model) to explain your thinking? What are other possibilities?

What would happen if . . . ?

Can you describe an approach (strategy) you can use to solve this?

What do you need to do next?

Do you see any patterns or relationships that will help you solve this?

How does this relate to ...?

Can you make a prediction?

Why did you...?

What assumptions are you making?

What did you try that did not work?

Can the explanation be made clearer?

Responding (helps clarify and extend their thinking)

Tell me more.

Can you explain it in a different way?

Is there another possibility or strategy that would work?

Is there a more efficient strategy?

Help me understand this part...

Adapted from They're counting on us, California Mathematics Council, 1995