5.1.2 Decimals & Fractions
Overview
Standard 5.1.2 Essential Understandings
The study of rational numbers now includes decimal representations to millionths as well as fractions. Fifth graders extend their understanding of the base ten numeration system and place value concepts to include millionths. For example: Possible names for the number 0.0037 are: 37 ten thousandths; 3 thousandths + 7 ten thousandths and a possible name for the number 1.5 is 15 tenths.
Students determine .1 more/less, .01 more/less and .001 more/less than a given number. They are able to compare and order fractions and decimals and locate them on a number line.
Students develop an understanding of conversion between fractions and decimals. Work with equivalent fractions continues as students encounter fractions with denominators of 15, 16, 20, 25, 50 and 100. These understandings are used in solving real-world and mathematical situations.
All Standard Benchmarks
5.1.2.1
Read and write decimals using place value to describe decimals in terms of groups from millionths to millions.
5.1.2.2.
Find 0.1 more than a number and 0.1 less than a number. Find 0.01 more than a number and 0.01 less than a number. Find 0.001 more than a number and 0.001 less than a number.
5.1.2.3
Order fractions and decimals, including mixed numbers and improper fractions, and locate on a number line.
5.1.2.4
Recognize and generate equivalent decimals, fractions, mixed numbers and improper fractions in various contexts.
5.1.2.5
Round numbers to the nearest 0.1, 0.01 and 0.001.
5.1.2.1
Read and write decimals using place value to describe decimals in terms of groups from millionths to millions.
For example: Possible names for the number 0.0037 are: 37 ten thousandths; 3 thousandths + 7 ten thousandths a possible name for the number 1.5 is 15 tenths.
5.1.2.2.
Find 0.1 more than a number and 0.1 less than a number. Find 0.01 more than a number and 0.01 less than a number. Find 0.001 more than a number and 0.001 less than a number.
5.1.2.3
Order fractions and decimals, including mixed numbers and improper fractions, and locate on a number line.
5.1.2.4
Recognize and generate equivalent decimals, fractions, mixed numbers and improper fractions in various contexts.
5.1.2.5
Round numbers to the nearest 0.1, 0.01 and 0.001.
For example: Fifth grade students used a calculator to find the mean of the monthly allowance in their class. The calculator display shows 25.80645161. Round this number to the nearest cent.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- convert between fraction and decimal representations of a number.
- know decimal names for common fractions such as ¼ as 0.25, ⅓ as 0.33 (repeating), ½ as 0.5 and ⅕ as 0.2 etc. to facilitate ordering, comparing fractions and decimals and converting to percents.
- find 0.1, 0.01, and 0.001 more or less than a number.
- locate and order fractions and decimals on a number line.
- order a set of numbers that includes fractions, decimals, and mixed numbers.
- locate fractions and decimals, including mixed numbers and improper fractions, on a number line.
- extend their understanding of the base ten numeration system and place value concepts to include millionths. For example: Possible names for the number 0.0037 are: 37 ten thousandths; 3 thousandths + 7 ten thousandths; a possible name for the number 1.5 is 15 tenths.
- round a number to the nearest 0.1, 0.01, 0.001.
- recognize and generate equivalent decimals, fractions, mixed numbers and improper fractions.
- easily translate between proper and improper fractions and mixed numbers.
Work from previous grades that supports this new learning includes:
- represent equivalent fractions using fraction models such as parts of a set, fraction circles, fraction strips, number lines and other manipulatives.
- use models to determine equivalent fractions.
- locate fractions on a number line.
- use models to order and compare whole numbers and fractions, including mixed numbers and improper fractions.
- use fraction models to add and subtract fractions with like denominators in real-world and mathematical situations.
- develop a rule for addition and subtraction of fractions with like denominators.
- read and write decimals with words and symbols.
- use place value to describe decimals in terms of thousands, hundreds, tens, ones, tenths, hundredths and thousandths.
- compare and order decimals and whole numbers using place value, a number line and models such as grids and base 10 blocks. For example, they could determine a fraction between ⅓ and ¼ or a decimal between .9 and .91.
- read and write tenths and hundredths in decimal and fraction notations using words and symbols.
- know the fraction and decimal equivalents for halves and fourths.
- round decimals to the nearest tenth.
- develop understanding of fraction equivalence. They recognize that two different fractions can be equal (e.g., 15/9 = 5/3), and they develop methods for generating and recognizing equivalent fractions.
- extend previous understandings about how fractions are built from unit fractions, composing fractions from unit fractions, decomposing fractions into unit fractions.
- estimate the relative size of fractions and decimals by using benchmarks such as 0, ½, and 1 and beyond.
NCTM Standards
Understand numbers, ways of representing numbers, relationships among numbers, and number systems
Grades 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
Understand the place value system.
5.NBT.1. Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
5.NBT.2. Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10.
5.NBT.3. Read, write, and compare decimals to thousandths.
5.NBT.3a.Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000).
5.NBT.3b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
5.NBT.4. Use place value understanding to round decimals to any place.
Misconceptions
Student Misconceptions and Common Errors
Students may think...
- the numerator and denominator are separate whole numbers.
- whole number relationships can be applied to fractions or decimals. For example, believing 0.26 is greater than 0.8 because 26 is greater than 8.
- whole numbers are always larger than fractions including mixed numbers.
- smaller is bigger with fractions - 1/7 is greater than 2/7- the smallest numerator is the largest piece or ½ is smaller than ⅓ because 2 is smaller than 3
- the difference between the denominator and the numerator indicates how close the fraction is to one. For example, ¾ and ⅔ are both one away from the whole so they are the same in size. Or, ½ and 5/7 - ½ must be larger since it is 1 away from the whole and 5/7 is 2 pieces away from the whole.
- decimals are just like whole numbers; everything you do with whole numbers you do with decimals.
- the more digits to the right of the decimal point the bigger the number.
- decimals are completely different from fractions.
Resources
Teacher Notes
- Students may need support in further development of previously studied concepts and skills.
- Equivalent fractions are created by multiplying by 1 (2/2, 3/3, 4/4), etc.
- Simplifying fractions requires dividing by 1 (2/2, 3/3, 4/4), etc.
- When comparing fractions only teaching students to find common denominators instead of building understanding using benchmark fraction of 0, 1/2 and 1 to estimate size reduces the opportunity for students to develop number sense.
- Annexing or placing zeroes to make the decimals being compared the same number of digits is misguided and does not allow students to focus on place value.
- Reading decimals correctly such as 0.26 as "twenty six hundredths" instead of "point two six" supports place value understanding.
- Use number lines to determine appropriate placement of decimals such as .9, .09, .19 etc.
- Careful use of language when modeling equivalent fractions is important. For example, using fractions strips to model changing ¾ to the equivalent fraction 6/8, focus the discussion on the change that occurs in the numerator and denominator when multiplying by 1 whole (2/2) vs. dividing each of the fourths into halves to get eighths. This creates confusion for students into believing the operation is division rather than multiplication.
- The symmetrical balance for whole numbers and decimals is the ones place not the decimal point.
- According to MCA III Test Specifications denominators are limited to 2, 3, 4, 5, 6, 8, 10, 12, 15, 16 and 20. When recognizing and generating equivalent fractions, decimals, mixed numbers and improper fractions denominators are limited to 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 25, 50 and 100.
- The Rational Number Project (Initial Fraction Ideas) 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 http://www.cehd.umn.edu/rationalnumberproject/rnp1-09.html
- The Rational Number Project (Fraction Operations and Initial Decimal Ideas) provides researched based strategies and lessons supporting conceptual understanding of fractions and decimals including connections to operations with fractions and decimals
Cramer, K., Wyberg, T., & Leavett, S. (2009). Rational number project: Fraction operations and initial decimal ideas. http://www.cehd.umn.edu/rationalnumberproject/rnp2.html
- 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)
NCTM Illuminations
- Lessons include: "Developing Fractions--the length model" and Developing Fractions--the set model." Activities include: Equivalent Fractions; Fraction Models; and the Fraction Game
- Decimals, Fractions & Percentages This game could be used when working with the equivalence of decimals, fractions, and percentages. -
Additional Instructional Resources
National Library of Virtual Manipulatives
Cramer, K., Behr, M., Post T., & Lesh, R. (2009). Rational number project: Initial fraction ideas
Cramer, K., Wyberg, T., & Leavett, S. (2009). Rational number project: Fraction operations and initial decimal 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.
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 3-5. Boston, MA: Pearson Education.
rational number: A number expressible in the form a/b or - a/b for some fraction a/b. The rational numbers include the integers.
numerator: The number that is written above the line in a fraction. It tells how many of the whole you have or how an y parts are being considered.
denominator: The number below the line in a fraction. It shows how many how many equal pieces the whole has been divided into.
mixed number: A number that has both a whole number part and a fractional part such as 2 ⅓. Mixed numbers represent values greater than 1.
improper fraction: A fraction in which the numerator is greater than the denominator such as 11/3. Improper fractions represent values greater than one.
"Vocabulary literally is the
key tool for thinking."
Ruby Payne
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. Theseopportunities 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 decimal understanding at the fifth grade level? How do student misconceptions interfere with mastery of these ideas?
How would you know a student understands the decimal system when using number from .0001 to .1?
What representations should a student be able to make for the number 365.4729 if they understand place value?
What experiences do students need in order to develop an understanding of rounding decimals to the nearest tenth, hundredth and thousandth?
When checking for student understanding of decimals, what should teachers
- listen for in student conversations?
- look for in student work?
- ask during classroom discussions?
Examine student work related to a place value task involving decimals 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.
What are the key ideas related to fraction understanding at the fifth grade level? How do student misconceptions interfere with mastery of these ideas?
What representations should a student be able to make for the fraction ______?
When checking for student understanding of fractions at the fifth grade level, 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 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.
What is meant by equivalent representations? How can teachers help students understand equivalent representations?
How can teachers assess student learning related to these benchmarks?
How are these benchmarks related to other benchmarks at the fifth 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. (2^{nd} 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.
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.
Reeves, D. (2007). Ahead of the curve: The power of assessment to transform teaching and learning. Indiana: Solution Tree Press.
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.
Wickett, M., & Burns, M. (2003). Teaching arithmetic: Lessons for extending division, grades 4-5. Sausalito, CA: Math Solutions.
Assessment
- Which number has a 5 in the ten thousandths place?
A. 0.20815
B. 0.30256
C. 0.40571
D. 0.50098
Solution : B 0.30256
Benchmark 5.1.2.1
MCA III Item Sampler
- Johan's race time was 45.03 seconds. Kyle's race time was 0.1 second less than Johan's time. What was Kyle's race time?
A. 44.03 seconds
B. 44.93 seconds
C. 45.13 seconds
D. 45.14 seconds
Solution: B 44.93. seconds
Benchmark 5.1.2.2
MCA III Item Sampler
- What is 0.45831 rounded to the nearest thousandth?
A. 0.45
B. 0.458
C. 0.459
D. 0.4583
Solution: B 0.458
Benchmark 5.1.2.5
MCA III Item Sampler
- Five points are shown on a number line.
Solution: B. K and L
Benchmark: 5.1.2.3
MCA III Item Sampler
- Lydia used 1/25 of her notebook paper. What decimal amount did she use?
A. 0.04
B. 0.4
C. 1.25
D. 2.5
Solution: A 0.04
Benchmark 5.1.2.4
MCA III Item Sampler
- Write a fraction, with an unlike denominator, that is larger than 2/7
Solution: Will vary.
Benchmark: 5.1.2.3
- Represent the fraction 8/6 in two other ways.
Solution: 1 2/6, 1 ⅓, 1.333
Benchmark: 5.1.2.4
- Write the decimal representation for the following: ⅓ 2/10 ⅜ 5/20 3/2
Solution: 0.33 0.2 0.375 1.5
Benchmark: 5.1.2.4
- Write the fraction representation for the following: 1.3 .666 .75 0.05 0.5
Solution: 1 3/10 2/3 3/4 1/20 1/2
Benchmark 5.1.2.4
Differentiation
- Structure consistent computational fluency activities utilizing physical models such as number lines and base ten blocks to help reconstruct multiplication/division facts when needed.
- Actively engage students in learning situations that focus on both concept and skill development (Place value millions to millionths) Provide explicit systematic instruction that includes opportunities for students to ask and answer questions and think aloud when making decisions while solving problems. Be sure that students understand the place value symmetry for whole numbers and decimals is one and not the decimal point.
- Instructional settings should include direct instruction work before and after the mathematics lesson such as I Do (teacher demonstrates), You do, (student models) or vocabulary instruction; whole group (students receive core instruction with classmates in regular classroom; and small group situations (such as partner work) that are well structured and have clear expectations. Make use of technology as appropriate.
- Use vocabulary graphic organizers such as the Frayer model (see below) to emphasize vocabulary words such as rational numbers, numerator, denominator, mixed number, and improper fractions.
- Carefully connect prior knowledge (place value thousands to thousandths to new learning of large numbers (millions to millionths) to read, write, compare and round decimals.
- Carefully connect prior knowledge (fractions, fraction benchmarks, and fraction models) to compare fractions with unlike denominators.
- Pose meaningful problems set in familiar situations.
- Incorporate visual models such as the number lines, fraction bars, and decimal grids.
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.
Additional Resources
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
Cramer, K., Wyberg, T., & Leavett, S. (2009). Rational number project: Fraction operations and initial decimal 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 3-5. Boston, MA: Pearson Education.
Teachers need to demonstrate and model the use of manipulatives (place value blocks) or representations such as bar and area models when connecting language and concepts as students work with fractions and decimals.
- 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 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 __________ is the same as the decimal __________________. |
The decimal __________ is the same as the fraction _________________. |
The decimal _____________ means ___________________________________. |
The fraction _____________ means ___________________________________. |
- 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.
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 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.
Parents/Admin
Administrative/Peer Classroom Observation
Students are... | Teachers are... |
explaining thinking for ordering fractions and decimals. | asking clarifying questions which illustrate student thinking. Helping students use benchmark numbers as referents when comparing and ordering fractions and decimals. |
finding equivalent representations of fractions and decimals. | providing a variety of models of fractions and decimals as they develop conceptual and procedural understanding of equivalent fractions and decimals. |
using appropriate mathematics vocabulary. | developing vocabulary throughout instruction. |
finding .001 more/less, .01 more/less and .1 more/less than a number. | providing representations for finding .001 more/less, .01 more/less and .1 more/less than a number. |
rounding to the nearest .1 .01, and .001. | providing number lines with appropriate scales as representations for rounding. |
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.
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.
Additional Resources
For Mathematics Coaches
Chapin, S. and Johnson, A. (2006). Math matters: Understanding the math you teach: Grades k-8. (2nd ed.). 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. (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.
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.
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