2.3.2 Length
Understand the relationship between the size of the unit of measurement and the number of units needed to measure the length of an object.
For example: It will take more paper clips than whiteboard markers to measure the length of a table.
Demonstrate an understanding of the relationship between length and the numbers on a ruler by using a ruler to measure lengths to the nearest centimeter or inch.
For example: Draw a line segment that is 3 inches long.
Overview
Standard 2.3.2 Essential Understandings
Second graders understand linear measure as an iteration of units and use rulers with that understanding. They understand the need for equal-length units, the use of standard units of measure (centimeter and inch), and relationship between the size of a unit and the number of units needed to measure length. They are able to use a ruler to measure length to the nearest inch and centimeter.
All Standard Benchmarks
2.3.2.1
Understand the relationship between the size of the unit of measurement and the number of units needed to measure the length of an object.
2.3.2.2
Demonstrate an understanding of the relationship between length and the numbers on a ruler by using a ruler to measure lengths to the nearest centimeter or inch.
2.3.2.1
Understand the relationship between the size of the unit of measurement and the number of units needed to measure the length of an object.
2.3.2.2
Demonstrate an understanding of the relationship between length and the numbers on a ruler by using a ruler to measure lengths to the nearest centimeter or inch.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- understand that the size of the unit impacts the number of units needed to measure the length of an object.
- understand the need for standard units and tools to measure length.
- understand the relationship between the numbers on a ruler and length
- accurately measure lengths to the nearest centimeter and inch using a ruler.
Work from previous grades that supports this new learning includes:
- Measure the length, size, and weight of an object using direct comparison.
- Measure the length of an object in terms of multiple copies of another object. For example: Measure a side of a table by placing paper clips end-to-end and counting the paper clips.
NCTM Standards
Understand measurable attributes of objects and the units, systems, and processes of measurement.
Pre-K - 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.
Pre-K - 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;
- use tools to measure;
- develop common referents for measures to make comparisons and estimates. (NCTM, 2000)
Common Core State Standards
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 the length difference in terms of a standard length unit.
Relate addition and subtraction to length.
- 2.MD.5. Use 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.
- 2.MD.6. Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, and represent whole-number sums and differences within 100 on a number line diagram.
Misconceptions
Student Misconceptions and Common Errors
Students may think...
- when using a ruler, they should start measuring by placing the end of the object to be measured on the "1" of the ruler instead of the "0."
- when using a ruler, they should start measuring by placing the end of the object to be measured at the end of the ruler instead of where the measuring units start.
- inches and centimeters are interchangeable.
- the length of an object changes as the number of units needed to measure its length changes.
- just a "number" describes the length of an object. The unit is not important.
Vignette
In the Classroom
This vignette takes place in a second grade classroom as they are well into their learning about measuring length. They have developed their vocabulary relating to length. Students realize the need for a standard unit. They've learned the correct way to use tools to measure objects. Their experiences with measuring objects have helped them develop a benchmark or a familiar object that is one centimeter and one that is one inch. They know to think of their benchmarks to estimate the length of objects. This activity will deepen their understanding of measurement and its practical uses.
Mrs. S.: Today we'll do a fun activity to practice our measuring skills. Our goal is to estimate and then accurately measure the length of things in our classroom. What does it mean to measure length?
Connor: It's what you do to find out how long or tall something is.
Jordyn: You can find out how many units a thing is, like how many inches.
Mrs. S.: Great ideas! Anything else?
Taylor: I like to use a ruler to measure.
Mrs. S.: Great! Using tools to measure length is very helpful. What are some
tools we can use?
Sarah: You can use a ruler with inches and centimeters.
Tucker: A yardstick would work.
Cade: Metersticks are a tool to measure.
Kadence: My mom uses a tape measure to measure around my waist and wrists when she sews my clothes.
Isabelle: The paper ruler on my name tag on my desk can be a tool.
Mrs. S.: Those are all tools to measure length, very good! Today we will be using a ruler to measure length. Do you have any helpful pointers for using a ruler?
Summer: Put the thing you are measuring right next to the ruler.
Luke: Line up one end of the object with the "0" on the left side of the ruler.
Carter: Look to the end of the object and find the closest inch mark on the ruler.
Mrs. S.: Right! You've given some great tips for using a ruler. We have learned about two different measurement systems. Can anyone tell us what they are?
Elle: The U.S. and Metric Systems.
Mrs. S.: Both systems are important to learn to use for measuring. What are some units to measure length in the U.S. Customary System?
Devin: Inches and feet.
Becca: A yard.
Parmanand: Miles.
Mrs. S.: Correct! What about the units in the Metric System? Can you name units to measure length?
Dominic: Meters.
Jenessia: Centimeters and decimeters.
Mrs. S.: You are right! It is important to learn both the U.S. Customary System and the Metric System as both are used in our world. Today we will use both to estimate and measure the length of things in our classroom. I'd like my materials helpers to pass out rulers to everyone. Since there are twenty students and we need partners, we'll have ten sets of partners. Let's number off, each saying the next counting number from 1 - 10 and then beginning with 1 - 10 again. Please remember your number. I'd like the ones to become partners, twos, and so on. Find your partner and stand next to him or her with your ruler.
Mrs. S.: (When partnerships are ready.) Your task is to search the room for an item that is 12 centimeters long. Look at things around the room. First estimate how many centimeters long an object is. When you think you found one that is about 12 centimeters, measure to find out. Be sure both you and your partner measure the object and agree that it is about 12 centimeters. Continue to search for more items that are 12 centimeters long, until I ring my chime. You will have 5 minutes. You may begin.
Mrs. S. ensures the partnerships are sharing the responsibility and both are engaged in learning and discussion. As students search and measure, she monitors their choice of objects to measure. She assesses whether the object they are trying is a reasonable estimate of 12 centimeters. She checks their use of their rulers for accurate measuring techniques. Mrs. S. steps in and assists when necessary. She also questions students to enrich their thinking about measuring.
Mrs. S.: Why did you choose this object? (to Jaelyn and Isabelle)
Jaelyn: We thought the tissue box might be 12 centimeters.
Isabelle: It is long and easy to measure.
Mrs. S.: How long is 12 centimeters?
Jaelyn: It's right here on my ruler.
Isabelle: It's 12 of these (points to the space between the 0 and 1 centimeter marks on her ruler).
Mrs. S.; Looking at the 12 centimeters on your ruler, is this tissue box a reasonable estimate of 12 centimeters?
Isabelle: No, it's almost double that.
Jaelyn: This tissue box is a lot longer. We need to find something shorter.
Mrs. S.: Good thinking, girls! Use your good estimation skills to help you find an object.
Mrs. S.: I noticed you measured the length of your pencil box and it was
more than 12 centimeters. Is there another length you could measure
on the pencil box? (to Brian and Desia)
Brian: We could measure this part. Brian points to the side representing the width of the pencil box.
Desia: Yeah, it might be 12 centimeters!
Brian: We could measure it this way (lays his centimeter ruler along the side of his pencil box).
Mrs. S.: Yes! You could measure its width in centimeters. Objects have many parts that can be measured.
Mrs. S.: I can see you have been working hard to find an object that is 12 centimeters long (to Connor and Jordyn). Did you find one?
Connor and Jordyn (frustrated): No.
Mrs. S.: What did you try that did not work?
Connor: My pencil and scissors.
Jordyn: The tape and stapler on your desk.
Connor: We tried a marker and a glue bottle, too.
Mrs. S.: You are determined partners! I can see you are on the right track.
You have tried many items that are very close to 12 centimeters.
Why do you think it's so hard to find one?
Jordyn: Because centimeters are so small.
Connor : It's hard to find one because the centimeters are so close and the numbers change quickly.
Mrs. S.: That's right! It does make it challenging, but I know you two will use your good estimation skills and just keeping trying. You'll find one.
Mrs. S.: Can you explain to me how you chose this clipboard? (to Sarah and Cade)
Sarah: We measured this long side and it was 12 centimeters.
Mrs. S.: Show me how you measured it.
Cade: We put the ruler along the side and lined the zero up with the end. Then we looked to the other end of the clipboard and it is on the 12 mark.
Mrs. S.: Which unit did you use to measure?
Sarah: Oh, no! That's right, we need to find a thing that's 12 centimeters!
Cade: We found 12 inches, oops!
Mrs. S.: It is really important to think about the unit when measuring. It's like saying 12 hours instead of 12 minutes when talking about time. The size of the unit changes the number of units needed. Show me 12 centimeters on your ruler.
Cade and Sarah: This is 12 centimeters.
Mrs. S.: Compare that to the side of the clipboard. What do you notice?
Sarah: Twelve centimeters is much less than the side of the clipboard.
Cade: Yeah, 12 centimeters is way shorter than 12 inches. We need to look for shorter things.
Mrs. S.: Excellent! Keep thinking about centimeters as you measure your things.
Mrs. S. brings the whole group together again for a discussion and sharing of the objects found to be 12 centimeters long. She then gives each partnership a different measurement on slips of paper: one inch, three inches, six inches, nine inches, twelve inches and one centimeter, 5 centimeters, 10 centimeters, 20 centimeters, and 30 centimeters. The pairs are asked to search the room for items that fit their measurement and collect them. After 15 - 20 minutes, she calls the students together to share their items. Pairs bring their items up in order from the shortest to longest measurements in both inches and centimeters and set them on a table with their slip indicating their measurement. Mrs. S. leads a discussion about what pairs noticed about the items and what they learned about estimating and measuring lengths today.
Resources
Teacher Notes
Students may need support in further development of previously studied concepts and skills.
"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.
- Demonstrate how to align the 0, on or near the left end of the ruler, with the left end of the object or picture being measured.
- Distinguishing between inches and centimeters can be challenging for some students. Providing concrete examples of inches and centimeters helps build meaning for these concepts. The unit cubes in base ten blocks are 1 cm long. The rods are 10 cm long.
- Label objects in the classroom with their measurements for students to use as concrete examples.
- Students should record their results with both the number and the unit of measurement. Second graders should also know the abbreviations for inch (in.) and centimeter (cm).
- Students should refer to their measurements with phrases such as close to ___ and about___ when they talk about specific measurements. This is because a measurement is always an approximation, it is never exact. No matter how precise the measurement; it can always be measured more accurately if smaller units are used.
- 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?
Reflecting about the Solution
How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?
How can you convince me your answer makes sense?
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.
As People Get Older They Get Taller
In this two-lesson unit, students compare the heights of friends and classmates at different ages. Through the course of the lessons, students are exposed to algebra, measurement, and data analysis concepts. A major theme of the unit is analyzing change.
Students often view linear measurement as a procedure in which a number is simply read off a ruler. The goal of this lesson is to have students gain experience in linear measurement by using a variety of measuring instruments to measure the heights of classmates, to discover the error inherent in measurement, and to search for patterns in data that are represented on a table. In this lesson, students compare results of measuring the same height using different methods, and discuss measurement error. They measure the heights of classmates and the heights of older students in their school, and construct a table of height and age data. The lesson is also designed to serve as a springboard for a second lesson in which students relate measurement to algebra and data analysis concepts.
A key goal for instruction on algebra at the elementary level is to analyze change, and to understand how change in one variable can relate to change in a second variable. The goal of this lesson is for students to explore how changes in students' ages relate to changes in their heights.
Additional Instructional Activities
- Use the book, How Big Is a Foot? by Rolf Myller, as a springboard to the need for a standard measure. In the book, the king wants to have his helper make a bed for his wife for her birthday. No one has seen a bed and they don't know how big to make it. So the king measured with his BIG feet around the queen and told the helper how big he wants it. The helper makes it with his LITTLE feet. The bed ends up being too small and they figure out they have to use the same size foot to measure.
Additional Instructional Resources
Dacey, L., Cavanagh, M., Findell, C. R., Greenes, C., Jensen Sheffield, 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
"Vocabulary literally is the key tool for thinking."
Ruby Payne
centimeter: unit of length in the metric system
inch: a unit of length in the US customary system
ruler: tool to measure length.
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 Use: Multiple and varied opportunities to use the words in context. These 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.
Additional Resources for Vocabulary Development
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.
Reflection - Critical Questions regarding the teaching and learning of these benchmarks
What are the key ideas related to measuring length at the second grade level? How do student misconceptions interfere with mastery of these ideas?
Why is the use of non-standard units so important in the conceptual understanding of measurement?
What experiences do students need in order to measure length successfully?
What common errors do second graders make when measuring length?
When checking for student understanding of measuring length, what should teachers
- listen for in student conversations?
- look for in student work?
- ask during classroom discussions?
Examine student work related to a task involving measuring length. 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 second 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 grade 2: 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.
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, Marilyn. (2007). About teaching mathematics: A k-8 resource (3rd ed.). Sausalito, CA: Math Solutions Publications.
Burns, M. (Ed). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.
Caldera, C. (2005). Houghton Mifflin math and English language learners. Boston, MA: Houghton Mifflin Company.
Carpenter, T., Fennema, E., Franke, M., Levi, L., & Empson, S. (1999). Children's mathematics cognitively guided instruction. Portsmouth, NH: Heinemann.
Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
Chapin, S., & Johnson, A. (2006). Math matters: Understanding the math you teach, grades K-8. (2nd ed.). 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., & 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.
Dougherty, B., Flores, A., Louis, E., & Sophian, C. (2010). Developing essential understanding of number & numeration pre-k-grade 2. Reston, VA: National Council of Teachers of Mathematics.
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 grade 2 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.
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.
Murray, M. (2004). Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.
Murray, M., & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k-8. Portsmouth, NH: Heinemann.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies grades K-5. Sausalito. CA: Math Solutions.
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. A., & Lovin, L. H. (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
- Which object do you need the most of to measure the length of the paper?
A. 
B.
C. 
D.
Solution: B
Benchmark: 2.3.2.1
- Measure the length of the twig to the nearest inch.
- Solution: 4 inches
- Benchmark: 2.3.2.2
- Measure the length of a new pencil using centimeters and large paper clips.
Solution: Correctly measures the length of the pencil using large paper clips
and centimeters
Benchmark: 2.3.2.1, 2.3.2.2
- What is the length of the spoon to the nearest inch?
(Images from etc.usf.edu//clipart/sitemap.php)
A. about 5 inches
B. about 6 inches
C. about 7 inches
D. about 8 inches
Solution: A. about 5 inches
Benchmark: 2.3.2.2
- What is the length of the key to the nearest centimeter?
(Images from etc.usf.edu//clipart/sitemap.php)
A. 10 centimeters
B. 9 centimeters
C. 8 centimeters
D. 5 centimeters
Solution: B. 9 centimeters
Benchmark: 2.3.2.2
- Summer found her bookmark is 5 units long. Did Summer use inches or centimeters for her unit?
Solution: inches
Benchmark: 2.3.2.1, 2.3.2.2
Differentiation
When using standard units of measurement, struggling students will need to use objects that are exactly one inch in length to measure another object. After the object has been measured, students can lay an inch ruler next to the one inch objects and describe what they see. It is important to help students understand what a ruler represents. Unit blocks in base ten blocks are one centimeter in length. These can be used to represent the use of a centimeter ruler as the is repeated.
To help students develop a greater understanding of the units of measure and the measurement tools, they can make an inch/centimeter measuring sticks. Using 1 inch/centimeter strips of paper in two colors, students glue them onto tag board in alternating colors to create an inch/centimeter measuring stick. They can use the measuring stick to measure objects.
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Concrete - Representational - Abstract Instructional Approach
The Concrete-Representational-Abstract Instructional Approach (CRA) is a researched 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.
Additional Resources
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.
Understanding what it means to measure requires measuring with multiple copies of a single unit. In order to make the transition to a ruler, students need to measure with objects that are an inch long or a centimeter long. Rulers can be introduced and placed next to the units. Students can visibly see what ten inches/centimeters are when the ruler is next to ten one inch/centimeter units.
Reinforce measuring vocabulary words such as long and tall. Introduce the word long with clear instruction. To find how long something is, we find how far it is from one end to the other end. Model how to find how long a pencil is in centimeters. Share your thinking as you choose a centimeter ruler as your tool, align the end of the eraser with 0, and count the centimeters to find the total number. The pencil is about 12 centimeters long. Ask students to share a word that can describe the pencil. (long) To find how tall something is, we find how far it is from the bottom to the top. Model how to find how tall a desk is in inches. Share your thinking as you choose a tape measure with inches as your tool, align the base of the leg with 0, hold the tape measure up along the desk, and count the inches to find the total number. This desk is about 23 inches tall. Invite students to share a word that can describe the desk. (tall)
Word banks need to be part of the student learning environment in every mathematics unit of study. Refer to these throughout instruction.
Use vocabulary graphic organizers such as the Frayer model (see below) to emphasize vocabulary words such as length, ruler, inch, centimeter.
- 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:
|
I measured a _________ using a _________ ruler. It is _____________ long. |
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I can use these tools to measure__________________________________________. |
- 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.
Additional ELL Resources
Bresser, R., Melanese, K., & Sphar, C. (2008). Supporting English language learners in math class, grades k-2. Sausalito, CA: Math Solutions Publications.
Students should focus on measuring the length of objects that are longer than the ruler. How do they deal with having to move the ruler?
Ask students to measure with a broken ruler--a ruler that does not have an end.
How do they account for not having the initial units on a ruler?
Additional Resources
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.
Parents/Admin
Administrative/Peer Classroom Observation
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Students are... |
Teachers are... |
|
using rulers to measure a variety of objects and recording the measurements using the proper unit labels. |
facilitating conversations comparing measurements of various objects and the most appropriate unit for a given object. |
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measuring a variety of objects using two different units and comparing/contrasting the measurements. |
facilitating conversations comparing and contrasting various units; e.g., Why is the number of inches less than the number of centimeters for the same object? Why is the number of centimeters greater than the number of decimeters? |
|
exploring measurement with nonstandard and standard units. |
consistently assessing students' learning for progress and for use in planning instruction. |
|
estimating the length of objects in centimeters and inches and then verifying by finding the actual measurement. |
Facilitating students' learning of estimation and checking the reasonableness of their measurements. For example, questioning if an eraser is closer to 1 cm long or 10 cm long. |
What should I look for in the mathematics classroom? (Adapted from SciMathMN,1997)
What are students doing?
- Working in groups to make conjectures and solve problems.
- Solving real-world problems, not just practicing a collection of isolated skills.
- Representing mathematical ideas using concrete materials, pictures and symbols. Students know how and when to use tools such as blocks, scales, calculators, and computers.
- Communicating mathematical ideas to one another through examples, demonstrations, models, drawing, and logical arguments.
- Recognizing and connecting mathematical ideas.
- Justifying their thinking and explaining different ways to solve a problem.
What are teachers doing?
- Making student thinking the cornerstone of the learning process. This involves helping students organize, record, represent, and communicate their thinking.
- Challenging students to think deeply about problems and encouraging a variety of approaches to a solution.
- Connecting new mathematical concepts to previously learned ideas.
- Providing a safe classroom environment where ideas are freely shared, discussed and analyzed.
- Selecting appropriate activities and materials to support the learning of every student.
- Working with other teachers to make connections between disciplines to show how math is related to other subjects.
- Using assessments to uncover student thinking in order to guide instruction and assess understanding.
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.
For Administrators:
Burns, M. (Edt).(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.
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.
Parent Resources
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.
Helping your child learn mathematics
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
Help Your Children Make Sense of Math
Ask the right questions
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?
Reflecting about the Solution
How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?
How can you convince me your answer makes sense?
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
Help your child to estimate and measure objects that he or she uses in everyday activities. For example, you might ask, "How long do you think this banana is?" Then, have your child measure its length in centimeters and inches, comparing it to their estimate.
Enjoy stories such as How Big Is a Foot? by Rolf Myller, How Tall, How Short, How Far Away? by David A. Adler, or Inch by Inch by Leo Lionni.