5.1.3 Addition & Subtraction with Fractions
Overview
Standard 5.1.3 Essential Understandings
Fifth graders develop an understanding of and fluency with addition and subtraction of mixed numbers and fractions with unlike denominators. They expand their understandings of equivalent fractions and addition and subtraction of fractions with like denominators. Students use models to explore the addition and subtraction of fractions with unlike denominators. They are able to connect models for adding and subtracting fractions to procedures for adding and subtracting fractions. Their prior work and current explorations with unlike denominators lead students to the important conclusion that common denominators are useful for adding and subtracting fractions. In addition, students estimate sums and differences with benchmark fractions to check reasonableness of solutions. For example, 3/7 + ⅔ does not equal 5/10 (adding numerator and denominator as if they were whole numbers) since ⅔ is greater than 5/10 = ½ before adding more to it.
Fifth graders develop an understanding of and fluency with addition and subtraction of decimals, including standard algorithms. Students are able to connect models for adding and subtracting decimals to procedures for adding and subtracting decimals. Students continue to expand their understanding of place value and number properties as they add and subtract decimals.
Fifth graders solve real-world and mathematical problems requiring addition and subtraction of decimals, fractions and mixed numbers These include problems involving measurement, geometry and data. They estimate sums and differences of decimals and fractions when assessing the reasonableness of results.
All Standard Benchmarks
5.1.3.1
Add and subtract decimals and fractions, using efficient and generalizable procedures, including standard algorithms.
5.1.3.2
Model addition and subtraction of fractions and decimals using a variety of representations.
5.1.3.3
Estimate sums and differences of decimals and fractions to assess the reasonableness of results.
5.1.3.4
Solve real-world and mathematical problems requiring addition and subtraction of decimals, fractions and mixed numbers, including those involving measurement, geometry and data.
5.1.3.1
Add and subtract decimals and fractions, using efficient and generalizable procedures, including standard algorithms.
5.1.3.2
Model addition and subtraction of fractions and decimals using a variety of representations.
5.1.3.3
Estimate sums and differences of decimals and fractions to assess the reasonableness of results.
5.1.3.4
Solve real-world and mathematical problems requiring addition and subtraction of decimals, fractions and mixed numbers, including those involving measurement, geometry and data.
What should students know and be able to do [at a mastery level] related to these benchmarks?
- Add and subtract decimals to the millionths place using multiple representations including standard algorithms.
- Add or subtract 0.1, 0.01, 0.001, 0.0001, 0.00001or 0.000001 to a given whole number or decimal.
- Estimate decimal sums and differences to check for reasonableness of results.
- Use the relationship of addition and subtraction to add and subtract fractions including standard algorithms.
- Model the addition and subtraction of fractions using a variety of representations.
- Estimate fraction sums and differences to check for reasonableness of results.
Work from previous grades that supports this new learning includes Understand unit fractions
- Understand the concept of and generate equivalent fractions
- Relate decimal fractions to fraction notation
- Understand that the size of a fractional part is relative to the size of the whole
- Understand the relationship between the size of the denominator and the size of the parts, i.e., as the denominator gets larger the size of the parts gets smaller.
- Fractions can be parts of a whole, parts of a set, points or distances on a number line.
- Understand that fractions and decimals are numbers and can be represented on a number line.
- Locate fractions, including improper fractions, and mixed numbers on a number line.
- Use models to show addition and subtraction of fractions with like denominators and generate a rule for addition and subtraction of fractions with like denominators
NCTM Standards
Compute fluently and make reasonable estimates
Grades 3 - 5 Expectations
- develop fluency with basic number combinations for multiplication and division and use these combinations to mentally compute related problems, such as 30 [math]\square[/math] 50;
- develop fluency in adding, subtracting, multiplying, and dividing whole numbers;
- develop and use strategies to estimate the results of whole-number computations and tojudge the reasonableness of such results;
- develop and use strategies to estimate computations involving fractions and decimals in situations relevant to students' experience;
- use visual models, benchmarks, and equivalent forms to add and subtract commonly used fractions and decimals;
- select appropriate methods and tools for computing with whole numbers from among mental computation, estimation, calculators, and paper and pencil according to the context and nature of the computation and use the selected method or tool.
Common Core State Standards
Use equivalent fractions as a strategy to add and subtract fractions.
5.NF.1. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)
5.NF.2. Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
5.NF.3. Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers; e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?
5.NF.4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
- 5.NF.4a. Interpret the product (a/b) × q as parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
- 5.NF.4b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
5.NF.5. Interpret multiplication as scaling (resizing).
- 5.NF.5a. Compare the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
- 5.NF.5b. Explain why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explain why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relate the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1.
5.NF.6. Solve real world problems involving multiplication of fractions and mixed numbers; e.g., by using visual fraction models or equations to represent the problem.
5.NF.7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
- 5.NF.7a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.
- 5.NF.7b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.
- 5.NF.7c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions; e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?
Misconceptions
Student Misconceptions and Common Errors
Students may think...
- you add or subtract numerators and denominators when adding or subtracting fractions. For example, 1/7 + 2/7 = 3/14 and ½ + ⅔ = ⅗.
- 1 2/7 - 4/7 = 12/7 - 4/7 instead of 1 2/7 - 4/7 = 7/7 + 2/7 - 4/7 = 9/7 - 4/7.
- that to estimate an answer they need to find the exact answer
- that when using the standard algorithm to add and subtract decimals, the digits are aligned from right to left (the same as whole numbers)
- the terms "greatest common factor" and "least common multiple" are interchangeable.
Resources
Teacher Notes
- Students may need suppoort in further development of previously studied and skills.
- While the idea of teaching "key words" might be appealing, it does not support student comprehension of word problems and can lead to incorrect solutions.
- Teachers need to address conceptual understanding of addition and subtraction of fractions as well as procedural fluency. Conceptual understanding is developed through the use of models. These models include fraction circles, fraction strips, number lines, etc.
- When using models, students need to record their thinking using numerical expressions as well as pictures and words.
- Student misconceptions are lessened when procedures are connected to physical models, drawings, and/or pictures.
- Students need to be encouraged to estimate sums and differences using benchmark fractions when assessing reasonableness of their solutions. For example, ½ + ⅔ is clearly greater than 1 because ⅔ is greater than ½.
- 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.
- 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.
- Modeling word problems is critical as students develop an understanding of operations and the relationships that exist between and among those operations. The bar model is an effective tool that can be used to represent the relationships in a variety of problem solving situations.For more information on the bar model, read Modeling Word Problems in the Mathematics Best Practice tab in the Resources section of the Minnesota Mathematics Framework.
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
Fraction Track Game
This applet allows students to individually practice working with relationships among fractions and ways of combining fractions. For a two person version of this applet see the Fraction Track E‑Example
Additional Instructional Resources
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.
Common denominator - fractions that have the same denominator are said to have common denominators.
Greatest common factor - (GCF) the largest factor that different numbers have in common:
given two numbers, 36 and 45,
the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36 and
the factors of 45 are 1, 3, 5, 9, 15, 45 and the largest
factor they have in common (or greatest common
factor) is 9, so 9 is the greatest common factor.
Least common multiple- (LCM) the least common multiple is the smallest number that is a common multiple of a set of numbers; the least common multiple of 2, 3 and 5 is 30 because it is the smallest number that they all go into evenly.
"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. 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 addition and subtraction of fractions, decimals and mixed numbers at the fifth grade level? How do student misconceptions interfere with mastery of these ideas?
How would you know a student understands addition and subtraction of fractions? What common errors do students make when adding and subtracting fractions?
How would you know a student understands addition and subtraction of decimals? What common errors do students make when adding and subtracting decimals?
How would you know a student understands addition and subtraction of mixed numbers? What common errors to students make when adding and subtracting mixed numbers?
What experiences do students need in order to develop an understanding of addition and subtraction of fractions, decimals, and mixed numbers?
What strategies might a student use when adding and subtracting fractions, decimals and mixed numbers?
Write real world problems involving the addition/subtraction of fractions, decimals, and mixed numbers.
When checking for student understanding of addition and subtraction of fractions, decimals and mixed numbers, 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 addition and/or subtraction of fractions, decimals and mixed numbers at the fifth grade level. 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 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.
Cramer, K., Behr, M., Post T., Lesh, R., (2009). Rational number project: initial fraction ideas http://www.cehd.umn.edu/rationalnumberproject/rnp1-09.html
Cramer, K., Wyberg, T., & Leavett, S. (2009). Rational number project: fraction operations and initial decimal ideas. http://www.cehd.umn.edu/rationalnumberproject/rnp2.html
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.
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
- Add 45.908 + 3.26
A. 46.234
B. 49.168
C. 49.24
D. 78.508
Solution: B 49.168
Benchmark 5.1.3.1
MCA III Item Sampler
- A fraction model is shown.
Solution: B.
Benchmark 5.1.3.2
MCA III Item Sampler
- Jill is 48 ⅝ inches tall. Lei is 47.5 inches tall. What is the difference in their heights?
A. 0.125 inch
B. 1.08 inches
C. 1.125 inches
D. 1.62 inches
Solution: C. 1.125 inches
Benchmark 5.1.3.4
MCA III Item Sampler
Performance Assessment
- Julie put a box on a shelf that is 96.4 centimeters long.
The box is 33.2 centimeters long.
What is the longest box she could put on the rest of the shelf?
Show all your work.
Solution: 63.2 centimeters long
Benchmark: 5.1.3.4
TIMMS released item
Differentiation
- While the idea of teaching "key words" might be appealing it does not support student comprehension of word problems and can lead to incorrect solutions.
- Struggling learners need further opportunities to develop conceptual understanding of addition and subtraction of fractions. Conceptual understanding is developed through the use of models. These models include fraction circles, fraction strips, number lines, etc. When using models, students need to record their thinking using numerical expressions as well as pictures and words.
- Misconceptions can be addressed by connecting procedures to physical models, drawings, and/or pictures.
- Use vocabulary graphic organizers such as the Frayer model (see vocabulary section) to emphasize words such as tenths to millionths, numerator, denominator, equivalent fractions, improper fractions, mixed number, like and unlike denominators.
- Incorporate visual models such as fraction circles, fraction strips or number lines to emphasize relationships between fractions.
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 http://www.cehd.umn.edu/rationalnumberproject/rnp1-09.html
Cramer, K., Wyberg, T., & Leavett, S. (2009). Rational number project: fraction operations and initial decimal ideas.
http://www.cehd.umn.edu/rationalnumberproject/rnp2.html
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.
- While the idea of teaching "key words" might be appealing it does not support student comprehension of word problems and can lead to incorrect solutions.
- Conceptual understanding is developed through the use of models. These models include fraction circles, fraction strips, number lines, etc. When using models, students need to record their thinking using numerical expressions as well as pictures and words.
- Misconceptions can be addressed by connecting procedures to physical models, drawings, and/or pictures.
- 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:
I need to find common denominators to add and subtract fractions because____________________. |
I know the answer is more than _____________ because _________________________. |
- When assessing the math skills of an ELL student it is important to determine if the student has difficulty with the math concept or with the language used to describe the concept and conceptual understanding.
Additional ELL Resources
Bresser, R., Melanese, K., & Sphar, C. (2008). Supporting English language learners in math class, grades 3-5. Sausalito, CA: Math Solutions Publications.
Cramer, K., Behr, M., Post T., Lesh, R., (2009). Rational number project: initial fraction ideas http://www.cehd.umn.edu/rationalnumberproject/rnp1-09.html
Cramer, K., Wyberg, T., & Leavett, S. (2009). Rational number project: fraction operations and initial decimal ideas.
http://www.cehd.umn.edu/rationalnumberproject/rnp2.html
- Students could develop a greater understanding of adding and subtracting fractions with unlike denominators.
- Students would be given specific constraints and asked to write real world problems using those constraints. For example, using 3 fractions with unlike denominators, write an addition word problem where the sum is ⅞.
- Write a word problem where the sum is a mixed number greater than 2.
- Using two mixed numbers with unlike denominators, write a word problem where the difference or comparison is less than 1.
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 http://www.cehd.umn.edu/rationalnumberproject/rnp1-09.html
Cramer, K., Wyberg, T., & Leavett, S. (2009). Rational number project: fraction operations and initial decimal ideas.
http://www.cehd.umn.edu/rationalnumberproject/rnp2.html
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... |
using models including number lines, fraction circles, fraction bars and pictures to add and subtract fractions.
| providing fraction models and using them to illustrate student thinking as they add and subtract fractions. |
using models including number lines, place value materials,and pictures to add and subtract decimals. | providing decimal models and using them to illustrate student thinking as they add and subtract decimals. |
explain strategies and justify solutions as they solve problems. | asking students for generalizations they see in solution strategies. How are these strategies alike/different? |
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.
Cramer, K., Behr, M., Post T., Lesh, R., (2009). Rational number project: initial fraction ideas http://www.cehd.umn.edu/rationalnumberproject/rnp1-09.html
Cramer, K., Wyberg, T., & Leavett, S. (2009). Rational number project: fraction operations and initial decimal ideas. http://www.cehd.umn.edu/rationalnumberproject/rnp2.html
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.
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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