# 3.1.3 Fractions

3
Subject:
Math
Strand:
Number & Operation
Standard 3.1.3

Understand meanings and uses of fractions in real-world and mathematical situations.

Benchmark: 3.1.3.1 Fraction Representation

Read and write fractions with words and symbols. Recognize that fractions can be used to represent parts of a whole, parts of a set, points on a number line, or distances on a number line.

For example: Parts of a shape ($\frac{3}{4}$ of a pie), parts of a set (3 out of 4 people), measurements ($\frac{3}{4}$ of an inch).

Benchmark: 3.1.3.2 Size of a Fractional Part

Understand that the size of a fractional part is relative to the size of the whole.

For example: One-half of a small pizza is smaller than one-half of a large pizza, but both represent one-half.

Benchmark: 3.1.3.3 Order & Compare Fractions

Order and compare unit fractions and fractions with like denominators by using models and an understanding of the concept of numerator and denominator.

## Overview

Big Ideas and Essential Understandings

Standard 3.1.3 Essential Understandings

Third graders understand, interpret and represent fractions with symbols and words. They know that fractions can be used to represent:

• parts of a whole or shape: ¾ of a pie and 3 one-fourths.
• parts of a set, 3 out of 4 people.
• points or distances on a number line, ¾ of an inch.

They represent fractions using set models, area models and linear models.

Third graders understand that the size of a fractional part is dependent upon the size of the whole: ½ of a small pizza is smaller than ½ of a large pizza.

They understand the concept of numerator and denominator and compare and order unit fractions (1/2, ⅓, ¼) and fractions with like denominators (2/6, 6/6, 4/6).

Benchmark Cluster

Benchmark Group A

3.1.3.1 Read and write fractions with words and symbols. Recognize that fractions can be used to represent parts of a whole, parts of a set, points on a number line, or distances on a number line. For example: Parts of a shape (3/4 of a pie), parts of a set (3 out of 4 people), and measurements (3/4 of an inch).

3.1.3.2 Understand that the size of a fractional part is relative to the size of the whole. For example: One-half of a small pizza is smaller than one-half of a large pizza, but both represent one-half.

3.1.3.3 Order and compare unit fractions and fractions with like denominators by using models and an understanding of the concept of numerator and denominator.

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

• read fractions written in word form and number form.
• identify fractions as parts of a whole, parts of a set and points/distances on a number line
• understand the meaning of the numerator and denominator.
• translate between concrete and symbolic representations of fractions
• demonstrate the size of a fraction in comparison to the whole using multiple fraction models, fraction circles, fraction bars, grids, number lines, etc. ( how many pieces of each size do you need to make a whole).
• understand that as the number of equal pieces of a whole increases the size of each piece decreases in size
• understand a fraction 1/b as the quantity formed by 1 part when the whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
• represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
• represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Students will recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
• compare and order unit fractions and fractions with like denominators

Work from previous grades that supports this new learning includes:

• Compare and order whole numbers up to 1000
• Understand greater than and less than
• Informal knowledge of a half
Correlations

NCTM Standards

Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

Grade 3 - 5 Expectations:

• understand the place-value structure of the base-ten number system and be able to represent and compare whole numbers and decimals;
• recognize equivalent representations for the same number and generate them by decomposing and composing numbers;
• develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and as divisions of whole numbers;
• use models, benchmarks, and equivalent forms to judge the size of fractions;
• recognize and generate equivalent forms of commonly used fractions, decimals, and percents;
• explore numbers less than 0 by extending the number line and through familiar applications;
• describe classes of numbers according to characteristics such as the nature of their factors.

Common Core State Standards

Develop understanding of fractions as numbers.

• 3.NF.1. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
• 3.NF.2. Understand a fraction as a number on the number line; represent fractions on a number line diagram
• 3.NF2a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
• 3.NF.2b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
• 3.NF.3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.
• 3.NF.3a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line.
• 3.NF.3b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model.
• 3.NF.3c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.
• 3.NF.3d. Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

## Misconceptions

Student Misconceptions

Students may think...

• any object divided into two parts, regardless of their size or area, is divided into halves. any object divided into three parts, regardless of their size or area, is divided into thirds, etc.

• ½ of different objects are the same size because ½ is ½, not realize the size of the whole determines the size of the parts.
• the terms numerator and denominator are interchangeable.
• the relative size of a whole isn't necessary to know to determine what size the fraction is (i.e. one-half of a small pizza is smaller than one-half of a large pizza.
• the fraction with the larger denominator as being larger, applying their whole number understanding to comparing and ordering fractions (for example: ⅛ is larger than ⅙).
• the number 1 cannot be represented as a fraction (i.e. 4/4 = 1 whole).
• a fraction model divided into eight equal parts and having three parts shaded is 3/5 instead of 3/8.

## Resources

Instructional Notes

Teacher Notes

• Students may need support in further development of previously studied concepts and skills.
• Use fraction models, including the following, to represent fractions:
• parts of whole-fractions circles, fraction strips
• parts of a set-no more than 12 objects in a set
• number lines
• Students need to understand the connection between the number line representation of a fraction and the parts of a whole representation.
• Research has shown that the circular whole is most meaningful in the conceptual development of fractional understanding for students.
• Teachers need to help students translate between the symbolic and concrete representations of fractions.
• Students need to interpret a fraction as parts of a whole as well as a number of unit fractions. For example, ⅔ as 2 parts out of 3 equal parts and as two ⅓'s (1/3 and 1/3).
• The Rational Number Project provides researched based strategies and lessons supporting conceptual understanding of fractions including connections to operations with fractions.
• Students find the location of fractions with denominators of 2 and 4 on a number line. The decision as to where to place a number on a number line reflects student thinking about the location of, as well as, the relative magnitude of numbers. Placing the same number on number lines with different intervals requires students to be flexible when thinking about the relative size of numbers.

For example, which point represents the location of 1/2 on the number line?

• Students need to experience fractions using many different physical models with connections to symbolic representations.
• Students should have fraction models available at all times. Student understanding is enhanced when students make the decisions regarding the use of fractions models.
• MCA III test Specifications limit denominators to 2, 3, 4, 6, and 8.
• 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)

Instructional Resources

NCTM Illuminations

• In this unit, students explore relationships among fractions through work with the length model. This early work with fraction relationships helps students make sense of basic fraction concepts and facilitates work with comparing and ordering fractions and working with equivalency.
• This lesson focuses on the relationship between parts and the whole. These relationships were developed earlier and require the students to consider the size or value of the same fraction when different "wholes" are compared (i.e., the value of x is relative to the whole; x of a small pie is not equivalent to x of a large pie). This lesson promotes problem solving and reasoning as the students compare similar fractions with different "wholes." Students develop communication skills as they work in pairs and share their understanding about the relationship between the value of a fraction and the whole.

Chapin, S., and Johnson, A. (2006). Math Matters, Understanding the Math You Teach, Grades K-8. (2nd ed.). Sausalito, CA: Math Solutions Press.

Craig, M., Eich, M., Morris, K., & Aguilar, K. (2005). Grades 3-5 mathematics assessment sampler. Virginia: NCTM, Inc.

Cramer, K., Behr, M., Post T., & Lesh, R. (2009). Rational number project: Initial fraction ideas

Duncan, N., Geer, C., Huinker, D., Leutzinger, L., Rathmell, E., & Thompson, C. (2007). Navigating through number and operations in grades 3-5. Reston, VA: National Council of Teachers of Mathematics.

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

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.

Van de Walle, J., & Lovin, L. (2006). Teaching student-centered mathematics grades 3-5. Boston, MA: Pearson Education.

New Vocabulary

New Vocabulary

"Vocabulary literally is the key tool for thinking."

Ruby Payne

fraction -        a symbol used to name part of a whole, part of a set, a number on a number line, or a measurement

numerator -    the number above the bar in a fraction. The part of the fraction that tells how many of the equal parts are being used.

denominator - the bottom number in a fraction. The part of a fraction that tells how many equal parts the whole is divided into.

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.

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.

Professional Learning Communities

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

What are the key ideas related to understanding fractions at the third grade level?  How do student misconceptions interfere with mastery of these ideas?

What experiences do students need in order to develop an understanding of fractions?

What representations should students experience when working with fractions?

What experiences do students need in order to compare and order fractions?

When checking for student understanding, 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 the understanding of fractions.  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 third grade level?

Professional Learning Community Resources

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 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.

Fosnot, C., & Dolk, M. (2002). Young mathematicians at work: Multiplication and division. Portsmouth, NH: Heinemann.

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.

Otto, A., Caldwell, J., Wallus Hancock, S., & Zbiek, R.(2011). Developing essential understanding of multiplication and division for teaching mathematics in grades 3 - 5. Reston, VA.: National Council of Teachers of Mathematics.

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. (2009). Focus in grade 3: Teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.

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, 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.

O'Connell, S., & SanGiovanni, J. (2011). Mastering the basic math facts multiplication and division strategies, activities & interventions to move students beyond memorization. Portsmouth, NH: Heinemann

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.

Uittenbogaard, W., & Fosnot, C. (2008). Mini-lessons for early multiplication and division, Grades 3-4. Portsmouth, NH: Heinemann.

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.

Wickett, M., Ohanian, S., &  Burns, M. (2002). Teaching arithmetic: Lessons for introducing  division, Grades 3-4. Sausalito, CA: Math Solutions.

Yeatts, K., Battista, M., Mayberry, S., Thompson, D., & Zawojewski, J. (2004). Navigating through problem solving and reasoning in grade 3. Reston, VA: National Council of Teachers of Mathematics.

Zemelman, S., Daniels, H., & Hyde, A. (2005). Best practices: Today's standards for teaching and learning in America's schools. Portsmouth, NH: Heinemann.

## Assessment

• Amy, Elizabeth, Katie, Gretchen, and Deb love chocolate.  One afternoon they each had the same size chocolate bar.  This is what each girl ate:

Amy:               one-third of her chocolate bar

Elizabeth:        one-sixth of her chocolate bar

Katie:              one-fourth of her chocolate bar

Gretchen:        one-eighth of her chocolate bar

Deb:                one-half of her chocolate bar

Who ate the most chocolate?

Who ate the least chocolate?

Solution:  Deb ate the most chocolate, Elizabeth and Gretchen ate the least.

Benchmark: 3.1.3.2, 3.1.3.3

• Cory has 2 red crayons and 1 blue crayon.

What fraction of Cory's crayons is red?

A. 3/2

B. 2/3

C. 1/2

D. ⅓

Solution: B. ⅔

Benchmark: 3.1.3.1

MCA III Item Sampler

• Sarah and her sister have eight insects in a jar. Four of the insects are beetles. One is a firefly and three are crickets. What fraction of insects are beetles? What fraction of the insects are fireflies? What fraction of the insects are crickets?

Solution:          Beetles   4/8

Fireflies  ⅛

Crickets  3/8

Benchmark:3.1.3.1

MCA III Item Sampler

• Which point represents the location of 1/2 on the number line?

Solution:          1. B.

2. A.

Benchmark:     3.1.3.1

• Place the following fractions on the number line below:

2/4, ¾, ¼

Solution:

Benchmark: 3.1.3.3

## Differentiation

Struggling Learners

Struggling Learners

• In order to develop conceptual understanding of fractions students need to represent fractions in more than one way. These representations include:
• parts of whole--fractions circles, fraction strips
• parts of a set--no more than 12 objects in a set
• number lines
• Students should work with unit fractions before using fractions with like denominators.
• At the third grade level, denominators are limited to 2, 3, 4, 6, and 8.
• The Rational Number Project provides researched based strategies and lessons supporting conceptual understanding of fractions including connections to operations with fractions.

Cramer, K., Behr, M., Post T., & Lesh, R. (2009).  Rational number project: Initial fraction ideas.

Concrete - Representational - Abstract Instructional Approach

(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.

Cramer, K., Behr, M., Post T., & Lesh, R. (2009).  Rational number project: Initial fraction ideas.

Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades 3-5. 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.

Van de Walle, J. & Lovin, L. (2006). Teaching student-centered mathematics grades 3-5. Boston, MA: Pearson Education.

English Language Learners
• In order to develop conceptual understanding of fractions students need to represent fractions in more than one way. These representations include:
• parts of whole--fractions circles, fraction strips
• parts of a set--no more than 12 objects in a set
• number lines.
• The Rational Number Project provides researched based strategies and lessons supporting conceptual understanding of fractions including connections to operations with fractions.
• Word banks need to be part of the student learning environment in every mathematics unit of study.
• Word 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 fraction ______ means __________________________________.
 In the fraction _____, the numerator is _____and the denominator is ____.
 The fraction ____________ is more than the fraction ________________.
 The fraction ____________ is less than the fraction _________________.
• 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

Using physical models, students will represent fractions with unlike denominators. Using physical models, students will compare fractions with unlike denominators.

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 3-5.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

### Administrative/Peer Classroom Observation

 Students are ... Teachers are ... modeling fractions using parts of a whole, sets, and number lines. modeling fractions using area models, set models, and linear models. using manipulatives and number lines to compare/order fractions and to justify thinking. asking students to explain their thinking using the language of whole and part. understanding and using the terms numerator and denominator as they explain their thinking. using vocabulary of fractions, whole, set, number line, numerator and denominator throughout instruction.

What should I look for in the mathematics classroom?

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.

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

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

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

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

• Note: Cooking also involves the use of fractions in measurement situations.