4.1.2A Fractions
Represent equivalent fractions using fraction models such as parts of a set, fraction circles, fraction strips, number lines and other manipulatives. Use the 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.
For example: Locate $\frac{5}{3}$ and $1\frac{3}{4}$ on a number line and give a comparison statement about these two fractions, such as "$\frac{5}{3}$ is less than $1\frac{3}{4}$."
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.
Overview
Essential Understandings
Fourth graders expand their work with fractions to include representation of equivalent fractions. They use models to compare and order whole numbers and fractions, including improper fractions and mixed numbers. They are able to locate fractions on a number line. Fourth graders add and subtract fractions with like denominators and develop a rule for this action.
Fourth graders read, write and represent decimals in terms of thousands, hundreds, tens, ones, tenths, hundredths and thousandths. They use place value understanding, models, and number lines to compare and order decimals. They expand their understanding of rounding to include rounding of decimals to the nearest tenth.
Fourth graders use their knowledge of both fractions and decimals to read and write tenths and hundredths using both decimal and fraction notation. They know the decimal and fraction equivalents for halves and fourths.
All Standard Benchmarks
4.1.2.1 Represent equivalent fractions using fraction models such as parts of a set, fraction circles, fraction strips, number lines and other manipulatives. Use the models to determine equivalent fractions.
4.1.2.2 Locate fractions on a number line. Use models to order and compare whole numbers and fractions, including mixed numbers and improper fractions. For example: Locate on a number line and give a comparison statement about these two fractions, such as "... is less than ..."
4.1.2.3 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.
4.1.2.4 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. For example: Writing 362.45 is a shorter way of writing the sum: 3 hundreds + 6 tens + 2 ones + 4 tenths + 5 hundredths, which can also be written as: three hundred sixty-two and forty-five hundredths.
4.1.2.5 Compare and order decimals and whole numbers using place value, a number line and models such as grids and base 10 blocks.
4.1.2.6 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. For example: = 0.5 = 0.50 and = = 1.75, which can also be written as one and three-fourths or one and seventy-five hundredths.
4.1.2.7 Round decimals to the nearest tenth. For example: The number 0.36 rounded to the nearest tenth is 0.4.
Benchmark Group A
4.1.2.1 Represent equivalent fractions using fraction models such as parts of a set, fraction circles, fraction strips, number lines and other manipulatives. Use the models to determine equivalent fractions.
4.1.2.2 Locate fractions on a number line. Use models to order and compare whole numbers and fractions, including mixed numbers and improper fractions.
4.1.2.3 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.
What students should know and be able to do [at a mastery level] related to these benchmarks.
Students will be able to:
- use fraction models, including the following, to represent and determine equivalent fractions
- parts of whole--fractions circles, fraction strips
- parts of a set
- number lines
- use models to compare and order whole numbers, fractions, including mixed numbers and improper fractions.
-
place a variety of fractions (including mixed 1 1/2 and improper 3/2) and whole numbers accurately on a number line given pre-placed benchmarks. For example: Place 1/2, 3/4, 3/2, and 1 1/4 on a number line.
-
accurately add and subtract fractions with like denominators and describe the process for this computation.
Work from previous grades that supports this new learning
- know fractions can represent parts of a set, parts of a whole, a point on a number line as well as distance on a number line
- understand the concept of numerator and denominator
- understand that the size of a fractional part is relative to the size of the whole (a half of a small pizza is smaller than a half of a large pizza but both represent one-half)
- compare and order unit fractions
- compare and order fractions with like denominators
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 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.
- Compute fluently and make reasonable estimates
Grade 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$\square$50;
Common Core State Standards
Extend understanding of fraction equivalence and ordering.
4.NF.1. Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
4.NF.2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
4.NF.3. Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
- 4.NF.3a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
- 4.NF.3b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
- 4.NF.3c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
- 4.NF.3d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.
4.NF.4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
- 4.NF.4a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4).
- 4.NF.4b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)
- 4.NF.4c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?
Understand decimal notation for fractions, and compare decimal fractions.
4.NF.5. Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.2 For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.
4.NF.6. Use decimal notation for fractions with denominators 10 or 4.NF.7. Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.
Misconceptions
Student Misconceptions and Common Errors
Students may think:
- fractions cannot be written to represent more than one whole.
- all numerators have to be less than the denominator.
- mixed numbers do not represent improper fractions
- improper fractions do not represent mixed numbers
- one-half (or any given fraction) of a whole is always the same as one-half (or any given fraction) of a different whole.
- adding fractions means adding numerators and adding denominators.
○ For example: $\frac{a}{b}$ + 1/4= 2/8 when in reality it equals 2/4.
Using a model helps children clarify this misconception.
Vignette
In the Classroom
Students in this classroom have been working with fraction strips for four days. On the first day they made, labeled and explored with their new tool (fraction strips). On days two through four they learned different games that helped them find equivalent fractions, compare fractions, and allowed for adding and subtracting fractions with like denominators using the models.
In this vignette, two students are sharing their strategy for finding equivalent fractions, mixed numbers and improper fractions after playing "Choose Five." In Choose Five, pairs of students draw five fraction strips from the fraction kit envelope. Using these five pieces they are to find equivalent fractions as well as improper fractions and mixed numbers if the pieces chosen allow. Their job is to find two different equivalent fractions. *If the set of fractions chosen allows, they are to find an improper and mixed number. Students A and B were partners on this activity. They are the second of three sets of partners to share.
Student A: I won "rock, paper, scissors" so I chose 3 fraction pieces and B chose two. I pulled out two 1/4s and a 1/8, so you could say a 2/4 and a 1/8.
Student B: I pulled out the fractions 1/2 and 1/8. We laid them down in order from the smallest to the largest.
Student A: Then we looked for another fraction that could be laid down that would be exactly the same length (or size).
Student B: I thought we should try 1/8s because we had two of them and the set of 1/4s and I knew two 1/8s would cover 1/4.
Student A: So we needed to use all the 1/8s left from my set and then 3 more 1/8s from B's set. We needed a total of 10/8s. Two for each of the 1/8s we had...
Student B: Then two 1/8s for each of the 1/4s, so that was 4 more...
Student A: And then 4/8s to equal the 1/2. We had 10 1/8s in all.
Student B: The fraction we made was 10/8s and that's an improper fraction.
Teacher: You were able to make an improper fraction; were you able to make the mixed number?
Student A: Yeah, we were sorta lucky with our picked fractions. When you put them down in order it was really easy to see how it made a whole and 2/8s and that is the same as 1 and 2/8s.
Student B: Then we decided to use the fewest pieces we could and ended up with 1 and 1/4. That means that 1 and 2/8s is the same as 1 and ¼.
Student A: We messed around with it to see what other improper fractions we could have made. We didn't lay all pieces but we figured out since each 1/8 is the same as 2/16s we could have used 20/16s or...
Student B: 5/4s, since 2/8s make 1/4. I like that one the best...fewer pieces to mess with.
Several nods of agreements and "same here" come from the others in the class. The class continued as other pairs of students shared the fractions they made while playing "Choose Five."
Resources
Teacher Notes
- Students may need support in further development of previously studied concepts and skills.
- 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.
- Teacher modeling of fraction representations and operations supports student conceptual development and can be used to help correct student misconceptions.
- 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.
- Students need to construct a whole when given a fractional part. For example, if this is ¾ draw a whole.
● Students should locate fractions on number lines with varying intervals.
For example, where would the fraction 3/4 be located on the following number lines?
- 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.
Questioning
Good questions and good listening will help children make sense of the mathematics, build self-confidence and encourage mathematical thinking and communication. A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."
Getting Started
What do you need to find out?
What do you know now? How can you get the information? Where can you begin?
What terms do you understand/not understand?
What similar problems have you solved that would help?
While Working
How can you organize the information?
Can you make a drawing (model) to explain your thinking? What are other possibilities?
What would happen if . . . ?
Can you describe an approach (strategy) you can use to solve this?
What do you need to do next?
Do you see any patterns or relationships that will help you solve this?
How does this relate to...?
Why did you...?
What assumptions are you making?
Reflecting about the Solution
How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?
How can you convince me your answer makes sense?
What did you try that did not work? Has the question been answered?
Can the explanation be made clearer?
Responding (helps clarify and extend their thinking)
Tell me more.
Can you explain it in a different way?
Is there another possibility or strategy that would work?
Is there a more efficient strategy?
Help me understand this part ...
Adapted from They're Counting on Us, California Mathematics Council, 1995.
Instructional Resources
NCTM Illuminations
Activity name: Equivalent Fractions
Learning Objective: Students will create equivalent fractions by dividing and shading squares or circles, and match each fraction to its location on the number line.
Activity name: Fun with Fractions
Learning Overview: In this unit, students explore relationships among fractions through work with the set 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.
Other Web-based games:
Additional Instructional Resources
Burns, M. (2001). Teaching arithmetic: Lessons for introducing fractions, Grades 4-5. 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
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 deWalle, J., Karp, K., Bay-Williams, J. (2010). Elementary and middle school mathematics: teaching developmentally. (7th ed.) Boston, MA: Allyn & Bacon.
Van deWalle, J. & Lovin, L. (2006). Teaching student-centered mathematics grades 3-5. Boston, MA: Pearson Education.
New Vocabulary
equivalent fractions: Fractions are equivalent when they represent the same quantity or region.
mixed number: Refers to a number written as a whole number and a fraction (it is a number that lies between two consecutive whole numbers or two consecutive integers).
improper fractions: A fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number); for example, 3/2 or 5/3 are both considered "improper fractions."
"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 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 a word bank as the need arises. Students refer to a word bank which leads to greater understanding and application of words in context when communicating mathematical ideas.
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 students 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 fraction understanding at the fourth grade level? How do student misconceptions interfere with mastery of these ideas?
- What representations should a student be able to make for a given fraction?
- What errors do fourth graders make when working with equivalent fractions?
- 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 fourth grade level?
Professional Learning Community Resources
Cramer, K., Behr, M., Post T., Lesh, R., (2009). Rational number project: initial fraction ideas
The Rational Number Project provides researched based strategies and lessons supporting conceptual understanding of fractions including connections to operations with fractions.
Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K.. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.
Barnett-Clarke, C., Fisher, W., Marks, R., Ross, S., Charles, R., & Zbiek, R.. (2010). Developing essential understanding of rational numbers for teaching mathematics in grades 3 - 5. Reston, VA.: National Council of Teachers of Mathematics.
Chapin, S. and Johnson, A. (2006). Math matters: Understanding the math you teach, grades k-8, 2nd Edition. Sausalito, CA: Math Solutions Press.
Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). Classroom discussions: Using math talk to help students learn (Grades K-6). Sausalito, CA: Math Solutions.
Empson, S., & Levi, L. (2011). Extending children's mathematics: Fractions and decimals. Portsmouth, NH.: Heinemann.
Fosnot, C., & Dolk, M. (2002). Young mathematicians at work: Constructing fractions, decimals, and percents. Portsmouth, NH: Heinemann.
Hyde, A. (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.
McNamara, J., & Shaughnessy, M. (2010). Beyond pizzas & pies: 10 essential strategies for supporting fraction sense, grades 3-5. Sausalito, CA. Math Solutions Publications.
Sammons, L. (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
Schielack, J.. (2009). Focus in grade 4: 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.
Barnett-Clarke, C., Fisher, W., Marks, R., Ross, S., Charles, R., & Zbiek, R. (2010). Developing essential understanding of rational numbers for teaching mathematics in grades 3 - 5. Reston, VA.: National Council of Teachers of Mathematics.
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 3-5. Sausalito, CA: Math Solutions Publications.
Burns, M. (Eds). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.
Cavanagh, M. (2004). Math to know: 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 Edition. Sausalito, CA: Math Solutions Press.
Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). Classroom discussions: Using math talk to help students learn (Grades K-6). Sausalito, CA: Math Solutions.
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.
Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC.: National Academies Press.
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.
Empson, S., & Levi, L. (2011). Extending children's mathematics: Fractions and decimals. Portsmouth, NH: Heinemann.
Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course. 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.
Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.
Leinwand, S. (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH: Heinemann.
Lester, F. (2010).Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for school administrators. Reston, VA: National Council of Teachers of Mathematics.
McNamara, J., & Shaughnessy, M. (2010). Beyond pizzas & pies: 10 essential strategies for supporting fraction sense, grades 3-5. Sausalito, CA. Math Solutions Publications.
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.
Schielack, J., Charles, R., Clements, D., Duckett, P., Fennell, F., Lewandowski, S., Trevino, E., & Zbiek, R. M., (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: NCTM.
Sammons, L., (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
Schielack, J. (2009). Focus in grade 4: Teaching with curriculum focal points. 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.
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.
West, L., & Staub, F. (2003). Content focused coaching: Transforming Mathematics lessons. Portsmouth, NH: Heinemann.
Assessment
- Write or draw 3 fractions that are equivalent to 1/2.
Solution: Answers will vary. For example: 2/4, 3/6, 5/10
Benchmark: 4.1.2.1
2. Circle the fractions that represent 1/4.
.
Solution: The first, third and fifth figures.
Benchmark: 4.1.2.1
3.Mrs. Smith was planting a garden. She divided it into ten equal sections. In 1/5 of the sections were tomato plants, one section was lettuce, one section was cucumbers, one section was corn and the rest of the sections are flowers. Make a diagram of the garden and label it using fractional parts. What portion of the garden are flowers?
Solution: 5/10 or ½ of the garden are flowers
Benchmark: 4.1.2.3
4. James had one week to read a book for school. He divided the pages into 7 equal parts. He read one part each day. At the end of the fifth day, what fractional part of the reading was done? How do you know? (Use words, pictures and/or numbers to support your thinking.
Solution: 5/7 Explanations will vary.
Benchmark: 4.1.2.3
See MN MCA III sample problems: #8, 9, and 17
Differentiation
The Rational Number Project includes lessons which develop the concept of a fraction as well as lessons involving addition and subtraction of fractions. Physical models are used to support student learning: Cramer, K., Behr, M., Post T., & Lesh, R. (2009). Rational number project: Initial fraction ideas
Concrete - Representational - Abstract Instructional Approach
The Concrete-Representational-Abstract Instructional Approach (CRA) is a research-based instructional strategy based on three stages during the learning process that has proven effective in enhancing the mathematics performance of students who struggle with mathematics.The Concrete Stage is the doing stage. The concrete stage is the most critical in terms of developing conceptual understanding of mathematical skills and concepts. At this stage, teachers use manipulatives to model mathematical concepts. The physical act of touching and moving manipulatives enables students to experience the mathematical concept at a concrete level. Research shows that students who use concrete materials develop more precise and comprehensive mental representations, understand and apply mathematical concepts, and are more motivated and on-task. Manipulatives must be selected based upon connections to the mathematical concept and the students' developmental level.
The Representational Stage is the drawing stage. Mathematical concepts are represented using pictures or drawings of the manipulatives previously used at the Concrete Stage. Students move to this level after they have successfully used concrete materials to demonstrate conceptual understanding and solve problems. They are moving from a concrete level of understanding toward an abstract level of understanding when drawing or using pictures to represent their thinking. Students continue exploring the mathematical concept at this level while teachers are asking questions to elicit student thinking and understanding.
The Abstract Stage is the symbolic stage. Teachers model mathematical concepts using numbers and mathematical symbols. Operation symbols are used to represent addition, subtraction, multiplication and division. Some students may not make a clean transfer to this level. They will work with some symbols and some pictures as they build abstract understanding. Moving to the abstract level too quickly causes many student errors. Practice at the abstract level will not lead to increased understanding unless students have a foundation based upon concrete and pictorial representations.
Additional Resources
Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA: Corwin Press.
Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades 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.
English Language Learners
- 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.
- Word banks need to be part of the student learning environment in every mathematics unit of study. Refer to these throughout instruction.
- Use vocabulary graphic organizers such as the Frayer model (see below) to emphasize vocabulary words such as 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 ______ and the fraction _______ are equivalent fractions. |
|
These fractions are equivalent____________________. |
|
The fraction _________ is greater 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.
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.
Extending the Learning
Explore addition and subtraction of fractions with unlike denominators using physical models.
The Rational Number Project includes lessons using models to add and subtract fractions with unlike denominators: Cramer, K., Behr, M., Post T., & Lesh, R. (2009). Rational number project: Initial fraction ideas.
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: (descriptive list) |
Teachers are: (descriptive list) |
|
If building fraction strips, "look for"s are denoted with * *accurately identifying fractional parts of the whole while folding, cutting and labeling. |
*asking questions, modeling, checking for accuracy as students fold, cut and label. |
|
using fraction pieces (circles, strips or other pre-made or manufactured kits) to determine equivalent fractions for a half (then move on to other fractions, such as 3/4 or 2/6). |
notes equivalencies by keeping a list of those found by students on the board, asking students to "prove" how they know 2/4 cover 1/2 and why 4/6 doesn't work. |
|
comparing fractions (using models when necessary) to determine which is larger. |
assisting with models, checking for accuracy, clarifying misconceptions by having students "build" the fractions being compared. |
|
putting pre-determined fractions (including mixed and improper fractions) in order from least to greatest (alone or in pairs). |
offering a list of fractions (including mixed and improper) to students, leading a discussion on how to "build" mixed or improper fractions using models like pattern blocks, fraction strips, etc. monitoring students as they work. |
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 edition. Sausalito, CA: Math Solutions.
Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.
Felux, C., & Snowdy, P. (Eds.). (2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.
Sammons, L., (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.
For Administrators
Burns, M. (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
http://www2.ed.gov/parents/academic/help/math/part.html#p1
What should I look for in the mathematics program in my child's school? A Guide for Parents developed by SciMathMN.
http://www.scimathmn.org/sub/parents_mathclass.htm
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
- Baking and cooking with your child offers opportunities to have "real life" experience with fractions especially if you were to double a recipe or make half of a recipe.
- Websites:
http://www.kidsolr.com/math/fractions.html