Fractions: A ProblemSolving Approach


 Owen Wade
 11 months ago
 Views:
Transcription
1 Fractions: A ProblemSolving Approach DeAnn Huinker, University of WisconsinMilwaukee (Submitted for publication in the Wisconsin Teacher of Mathematics, March 2010.) "What's a fraction?" Students, unfortunately, too often reply, "That's when you have a line with a number on top and a number on the bottom" rather than associating it with some quantity in everyday life that can be represented mathematically with symbols. The connection between realworld experiences and symbols is essential in enabling students with the power to make sense of fractions. Other conceptual knowledge connections needed by students to develop number sense and operation sense for fractions are shown in figure 1. These include connections among realworld experiences, concrete models and diagrams, oral language, and symbols. Mathematical Concept Symbolic Representation Concrete/Pictorial Representation RealWorld Representation Figure 1. Conceptual Knowledge Connections Students who are able to make these connections have demonstrated lasting ability to flexibly use their mathematical knowledge, both conceptual and procedural, to solve word problems. Towsley (1989) found that fourth grade students who had established these connections reasoned with fractions as quantities and not as two whole numbers when solving word problems. Other educators and researchers have also noted the importance of helping students make connections among various mathematical representations (Cramer & Bezuk 1991; Ellerbruch & Payne 1978; Lesh, Post, & Behr 1987). Students without connections among realworld experiences, concrete models and diagrams, oral language, and symbols lack the power to make sense of mathematics and do not see its usefulness in the world around them. They often have no choice but to rely on immature strategies such as key words or guessing in their attempts to solve problems. An instructional program for fractions that utilized a problemsolving approach as it emphasized these connections to develop fraction number sense and operation sense was implemented with a class of fifth grade students in a large urban school system. This article discusses the intended outcomes, the instructional sequence and activities, and the assessment of student learning. Intended Instructional Outcomes What fraction knowledge do we want fifth grade students to develop? By the end of the instructional program, it was hoped that students would: 1. Approach fraction word problems with confidence. 2. Use a variety of problem solving strategies to solve fraction word problems, such as acting out the situation with paper strips or by drawing diagrams, using invented symbolic procedures on paper and pencil, and visualizing actions on objects (mental logical reasoning). Fractions: A Problem Solving Approach (Huinker, 2010) 1
2 3. Demonstrate fraction number sense and operation sense by: (a) making connections among realworld, concrete/pictorial, oral language, and symbolic representations for all four operations; (b) seeing relationships among all four operations; (c) acquiring insight into the effects of an operation on a pair of fractions or a fraction and a whole number; and (d) realizing that a specific amount can have many names. 4. Communicate their fraction knowledge using oral, written, concrete, pictorial, and symbolic methods. Instructional Sequence and Activities The instructional activities encouraged students to investigate fractions through informal explorations. The experiences focused on the use of paper strips as concrete models and problem situations. The paper strips were used throughout all lessons. Too often manipulatives are used for a day or two and are then quickly put aside. This is not adequate time for students to construct understanding. The students were also encouraged to come up with their own methods for solving problems and to communicate their reasoning by sharing their approaches with each other. The instructional sequence for the initial fraction work and for addition and subtraction was modified from that presented by Ellerbruch and Payne (1978), Payne and Towsley (1990), and Payne, Towsley and Huinker (1990). Instructional ideas for multiplication and division were modified from Cramer and Bezuk (1991), Ott, Snook and Gibson (1991), Pothier and Sawado (1990) and Sweetland (1984). Oral Language and Comparisons. Each student was given a fraction kit which consisted of ten strips that measured 2 inches by 8.5 inches. These were their paper candy bars. Several days were spent folding the paper into fractions strips for halves, fourths, eighths, thirds, sixths, twelfths, fifths, and tenths. The other two strips remained wholes. The students discovered various ways to fold the strips into equalsized pieces and shared their strategies with each other. Throughout the days, the students were asked to show and compare many quantities with their fraction strips as they informally dealt with proper and improper fractions and mixed numbers. For example, Show me 1 fourth, 2 fourths, 3 fourths, 4 fourths, 5 fourths, 8 fourths. The students then worked in pairs to compare fractions presented as verbal or written oral language, such as "Which would you rather have 5 eighths of a candy bar or 5 sixths of a candy bar?" No fraction symbols were used for several days of instruction in order to establish a stronger connection between the concept and concrete representation. Each student would select one of the two fractions that were to be compared, show it with their fraction strips, and then check with their partner to see who had the larger fraction. Students found fractions that were larger than, smaller than, and equal to one whole candy bar. They also found that different fractions named the same amount, or were equivalent. When students were able to confidently use oral language to display and compare fractions and mixed numbers, it was time to introduce fraction symbols. Fraction Symbols and Comparisons. All students had previously worked with fraction symbols, but their use was deliberately delayed until students demonstrated some understanding of fraction quantities. Students used written oral names, such as 5 sixths in their written work and then the symbol, 5, was introduced and discussed as a short way to write it. The meaning of the top number was 6 interpreted as the number of parts and the bottom number as the size of the parts. Students now used fraction symbols along with oral language to display and compare fractions. It was interesting that some students continued to use written oral language rather than the symbols or used both. One student explained that she did this because the words were less confusing than the symbols. In this transition to using fraction symbols, students used their fraction strips to engage in the activity, "Who has the largest fraction?" This is version of highest card that uses a deck of fraction symbol Fractions: A Problem Solving Approach (Huinker, 2010) 2
3 cards. In another activity students were challenged to find fractions that were larger than, smaller than, and equivalent to one whole candy bar and to write about their discoveries. One purpose of this activity was to have students consider fractions such as 3 thirds or 3 3 and 8 eighths or 8. It is rare that students 8 have discussions or even see fractions in which the top number and the bottom number are the same. The understanding that this is equivalent to one whole is important for students to discuss. This means they have enough of the same sized parts to make a whole candy bar. It is also rare for students to explore fractions greater than one whole during initial fraction instruction. This is unfortunate as it leads to misconceptions. Building from the oral language and concrete experiences, students can readily understand and explain that a fraction such as 8 means that you have 8 of the samesized parts, that you 5 have more than enough to make a whole candy bar, in fact you have enough to make one whole candy bar and then still have 3 fifths of another candy bar. To extend the ideas to fraction symbols not restricted to their fraction strips, students went hunting for equivalent fractions. First they found all the names for one half among their fraction strips and then students invented many more, such as , 80,, and. Next the students went hunting for other names for 1 third, 2 thirds, 1 fourth, and 3 fourths. Another activity which focused on equivalent fractions, as well as informally explored addition, challenged students to work in pairs to find and record different ways to build one whole. This was a favorite exploration of the students. Paula and Brandy worked together and find the ways shown in Figure 2. Figure 2. Finding Ways to Make One Whole Fractions: A Problem Solving Approach (Huinker, 2010) 3
4 Addition and Subtraction. More structured work with addition and subtraction began by posing word problems for the students to act out with their fraction strips. Both addition and subtraction problems with like fractions and related fractions were investigated on the first day. Beginning only with like fractions is a source of student misconceptions and interference with later learning. Students should experience like and related fractions interchangeably from the first day of fraction instruction. Then they learn that you do not have different rules but rather like fractions are only a special case of adding fractions. For example, "Let's pretend that Shawn had 1 half of a candy bar and then Latarre gave him 3 fourths of another candy bar. How much would Shawn have then?" Some students lined up the fractions, compared them with a whole, and realized that the answer was 1 whole candy bar and 1 fourth of another. Other students reasoned that 1 half is the same as 2 fourths, so they traded in the half for fourths which resulted in an answer of 5 fourths. Whereas other students said that they could not solve the problem because the pieces were not the same size. After listening to the thinking and reasoning of their classmates, these students gained insight and were eager to try the next problem. Number sentences using written oral language were used to record the results of each problem at this initial stage of development in adding and subtracting fractions. For example, the problem was written on the board as 1 half + 3 fourths. Then after doing some trading it was rewritten as 2 fourths + 3 fourths = 5 fourths = 1 whole and 1 fourth. This use of oral language promotes thinking of fractions as quantities and discourages students from merely adding numerators and denominators. When students were successful with trading in or renaming fractions to get same size pieces, a connection was made from the written oral language to the fraction symbols. The students began using a cross out technique as shown in Figure 3. To show that they had made a trade or renamed a fraction, they merely crossed it out and wrote the equivalent fraction next to it. Some of the students had remembered how they had been shown to write fraction addition problems vertically in the previous grade. The class talked about this and it was unanimous that it makes much more sense to just write the equation horizontally and to cross out the fraction when you rename it. Figure 3. Adding Fractions by Renaming Now that the students were familiar with fraction symbols and problem situations, they were asked to pose their own word problems as shown in Figure 3. Often times they were given a specific number sentence for which they were to pose a problem. This strengthens the connection from symbols to real life experiences. Once they had posed a problem, they would find the solution by using their fraction strips. These problems often became the basis for further exploration by other students in future lessons. The problems were also compiled into a fraction word problem book. Multiplication and Division. The instructional unit also included several lessons focused on the exploration of multiplication and division of fractions through problem situations. Word problems were Fractions: A Problem Solving Approach (Huinker, 2010) 4
5 posed for students to act out with their fraction strips, and then they shared their approaches and reasoning with each other. After a word problem was solved, the mathematical operation was identified and symbols were used to write a number sentence to record the relationships among the numbers in the problem. For example, it the story involved combining equal parts then a multiplication equation was written. If the story involved taking an amount and separating it into equal parts, then a division equation was written. Students also drew diagrams as a written record of their solutions as shown in Figure 4. If It is important to note that all problems were presented to the students in context. The increased accessibility to the mathematical ideas in that students could use their fraction strips to act out the situations. The symbols just became a written record of the story context and the solution. Figure 4. Student Solutions to Multiplication and Division Problem Situations Students interchangeably investigated multiplication of a whole number and a fraction, such as 3 x 2 3 ; division of a whole number by a fraction, ; division of a fraction by a fraction ; and division of a mixed number by a whole number, Students were also asked to pose their own 2 multiplication and division word problems which were compiled in a fraction word problem book that was put into the class library. Assessment of Student Learning Journals. Each student had a Fraction Journal which was written in daily. The students were sometimes asked to compare fractions and explain their reasoning in writing or with diagrams in their journals. The students pose their own word problem for a specific number sentence and wrote it in their journal, as well as drew diagrams or used symbols to show and explain how to solve the problem. At other times the students were just asked to write down something that they had learned that day. The journal allowed for ongoing assessment of student learning throughout the instructional program, as well as selfassessment by the students. For example, one day Mike wrote, I don't understand the mixed fraction and it s confusing me. So could you help me? Interviews. The students were individually interviewed before and after the instructional program. They were asked to compare fractions, perform fraction computations, and solve fraction word problems. All but one student initially compared fractions based on the size of the individual numbers. They thought 1 6 was larger than 1 2 because six was larger than two. Only one of the 28 students initially recalled that fractions needed a common denominator before adding or subtracting them. All the other students stated that the answer to was 4 6 and the answer to was 4 5. The students were allowed to use paper circles or paper strips in the initial interview, but only a few choose to do so with futile attempts. Fractions: A Problem Solving Approach (Huinker, 2010) 5
6 During the followup interviews, all students dealt with the fraction symbols as representing some quantity and did not merely manipulate the symbols without meaning. Students either used the fraction strips or drew diagrams to help them solve the problems or they reasoned through the solving of the problem by mentally visualizing the fractions. All students solved the comparison problems correctly, almost all successfully solved the addition and subtraction word problems, and many students also solved the multiplication and division problems. Paperandpencil Test. The students were given a paperandpencil pretest and posttest. These test results further substantiated the results of the interviews. Before instruction, almost all students added or subtracted both numerators and denominators and compared fractions based on the individual numbers. Whereas after instruction, only a few students resorted to meaningless symbol manipulations. This class of fifth grade students were immersed in fraction instruction for almost four weeks for about an hour each day. The results show that a focus on problem solving and making connections among representations build a strong and solid foundation for making sense of fractions. Conclusions These students developed intuitive quantitative understandings of fractions which they used to solve problems, as well as to pose problems. The role of contexts were essential for student learning. Problem contexts were used each and every day of instruction from initial concept development through division of fractions. The challenge of students posing their own word problems and selecting their own contexts was also a regular and essential part of instruction and learning. Students can experience success with fractions if they are given opportunities to investigate and explore connections among realworld experiences, concrete models and diagrams, oral language, and symbols, to communicate their findings, and to construct their understandings. Many of the students commented, I learned that fractions are easy. The two students who summed it up best were Wallace who stated, I learned that fractions are the best because it's important to use fractions all of our life and Shannon who said, I learned that you can do mostly anything with fractions. Resources Cramer, K., & Bezuk, N. (1991). Multiplication of fractions: Teaching for understanding. Arithmetic Teacher 39, Ellerbruch, L. W., & Payne, J. N. (1978). A Teaching sequence from initial fraction concepts through the addition of unlike fractions. In M. Suydam (Ed.), Developing computational skills (pp ). Reston, VA: National Council of Teachers of Mathematics. Lesh, R., Post, T., & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.), Problems of Representation in the Teaching and Learning of Mathematics (pp ). Hillsdale, NJ: Lawrence Erlbaum. Ott, J. M., Snook, D. L., & Gibson, D. L. (1991). Understanding partitive division of fractions. Arithmetic Teacher 39, Payne, J. N., & Towsley, A. E. (1990). Implications of NCTM's standards for teaching fractions and decimals. Arithmetic Teacher 37, Payne, J. N., Towsley, A. E., & Huinker, D. M. (1990). Fractions and decimals. In J. N. Payne (Ed.), Mathematics for the Young Child (pp ). Reston, VA: National Council of Teachers of Mathematics. Pothier, Y., & Sawado, D. (1990). Partitioning: An approach to fractions. Arithmetic Teacher 38, Sweetland, R. (1984). Understanding multiplication of fractions. Arithmetic Teacher 32, Towsley, A. (1989). The use of conceptual and procedural knowledge in the learning of concepts and multiplication of fractions in grades 4 and 5. Doctoral dissertation, University of Michigan. Fractions: A Problem Solving Approach (Huinker, 2010) 6
Grade 4  Module 5: Fraction Equivalence, Ordering, and Operations
Grade 4  Module 5: Fraction Equivalence, Ordering, and Operations Benchmark (standard or reference point by which something is measured) Common denominator (when two or more fractions have the same denominator)
More informationA Deeper Look at the Core of Effective Mathematics Teaching: Empowering Students Dr. DeAnn Huinker University of Wisconsin Milwaukee
A Deeper Look at the Core of Effective Mathematics Teaching: Empowering Students Dr. DeAnn Huinker University of Wisconsin Milwaukee huinker@uwm.edu Wisconsin Mathematics Council (WMC) Annual Conference
More informationFractions. Cavendish Community Primary School
Fractions Children in the Foundation Stage should be introduced to the concept of halves and quarters through play and practical activities in preparation for calculation at Key Stage One. Y Understand
More informationRational Number Project
Rational Number Project Fraction Operations & Initial Decimal Ideas 2009 Kathleen Cramer Terry Wyberg Seth Leavitt This material is based upon work supported by the National Science Foundation under Grant
More informationFourth Grade Math Pacing Guide Columbus County Schools
Time Objective (2003 Curriculum) DPI Strategies (2003 Curriculum) 13 1.01 Develop number sense for rational numbers from.01 to at least 100,000 a. Connect model, number P. 3839 D, F (1.02c) word, and
More information6 th Grade. 5 day Unit. Fraction Squares. Circle. Bars. By: Amy Nikiel
6 th Grade 5 day Unit By: Amy Nikiel Fraction Squares Fraction Circle Fraction Bars 2 Table of Contents Page 1. Objectives 3 2. NCTM Standards. 4 3. NYS Standards.. 5 4. Resources... 5 5. Materials 6 6.
More informationPerformance Assessment Task Cindy s Cats Grade 5. Common Core State Standards Math  Content Standards
Performance Assessment Task Cindy s Cats Grade 5 This task challenges a student to use knowledge of fractions to solve one and multistep problems with fractions. A student must show understanding of
More information1 ST GRADE COMMON CORE STANDARDS FOR SAXON MATH
1 ST GRADE COMMON CORE STANDARDS FOR SAXON MATH Calendar The following tables show the CCSS focus of The Meeting activities, which appear at the beginning of each numbered lesson and are taught daily,
More informationUNDERSTANDING FRACTIONS: WHAT HAPPENS BETWEEN KINDERGARTEN AND THE ARMY?
Understanding Fractions: What Happens Between Kindergarten And The Army? UNDERSTANDING FRACTIONS: WHAT HAPPENS BETWEEN KINDERGARTEN AND THE ARMY? Karim Noura Bayside Secondary College In this paper I will
More informationNumber Sense and Numeration, Grades 4 to 6
Number Sense and Numeration, Grades to 6 Volume 5 Fractions A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6 2006 Every effort has been made in this publication to identify mathematics
More informationMath, Grades 35 TEKS and TAKS Alignment
111.15. Mathematics, Grade 3. 111.16. Mathematics, Grade 4. 111.17. Mathematics, Grade 5. (a) Introduction. (1) Within a wellbalanced mathematics curriculum, the primary focal points are multiplying and
More informationDecimals in the Number System
Grade 5 Mathematics, Quarter 1, Unit 1.1 Decimals in the Number System Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Recognize place value relationships. In a multidigit
More informationComputation of Fractions
National Center on INTENSIVE INTERVENTION at American Institutes for Research Computation of Fractions 000 Thomas Jefferson Street, NW Washington, DC 0007 Email: NCII@air.org While permission to reprint
More informationGrade 1 Data Analysis: Additional Sample 1 How Did You Come to School Today?
Grade 1 Data Analysis: Additional Sample 1 How Did You Come to School Today? Context The students had previous experience with tallying, graphing, and orally explaining what they leaned from their graphs.
More informationPlanning Guide. Grade 6 Improper Fractions and Mixed Numbers. Number Specific Outcome 4
Mathematics Planning Guide Grade 6 Improper Fractions and Mixed Numbers Number Specific Outcome 4 This Planning Guide can be accessed online at: http://www.learnalberta.ca/content/mepg6/html/pg6_improperfractionsmixednumbers/index.html
More informationUnit 11 Fractions and decimals
Unit 11 Fractions and decimals Five daily lessons Year 4 Spring term (Key objectives in bold) Unit Objectives Year 4 Use fraction notation. Recognise simple fractions that are Page several parts of a whole;
More informationMathematics Instructional Cycle Guide
Mathematics Instructional Cycle Guide Fractions on the number line 3NF2a Created by Kelly Palaia, 2014 Connecticut Dream Team teacher 1 CT CORE STANDARDS This Instructional Cycle Guide relates to the following
More informationCommon Core State Standards (CCSS) Number and Operations Fractions (NF) in Math Expressions
hmhco.com BUILDING A New Standard OF Success Common Core State Standards (CCSS) Number and Operations Fractions (NF) in Math Expressions Dr. Karen Fuson, Program Author of Math Expressions Common Core
More informationPerformance Assessment Task Picking Fractions Grade 4. Common Core State Standards Math  Content Standards
Performance Assessment Task Picking Fractions Grade 4 The task challenges a student to demonstrate understanding of the concept of equivalent fractions. A student must understand how the number and size
More information1 st What is mathematics?
s 1 st What is mathematics? What is a number? How is math being discovered or invented? What should be learned in a math course? What should be learned about understanding numbers? What math knowledge
More informationThe development of number concepts is a major emphasis of mathematics instruction with
Calculators as Learning Tools for Young Children s Explorations of Number The development of number concepts is a major emphasis of mathematics instruction with young children. This development includes
More informationSupplemental Fraction Unit for Grade Five based on the Common Core Standards
Supplemental Fraction Unit for Grade Five based on the Common Core Standards SCLME (SC Leaders of Mathematics Education) recommends that district mathematics curriculum leaders support teachers with the
More informationComparing Fractions. The expectation that all students should use. The Roll Out Fractions Game: General Rules
The Roll Out Fractions Game: Comparing Fractions By Enrique Ortiz The expectation that all students should use models, benchmarks, and equivalent forms to judge the size of fractions is clearly stated
More informationNumber Talks. 1. Write an expression horizontally on the board (e.g., 16 x 25).
Number Talks Purposes: To develop computational fluency (accuracy, efficiency, flexibility) in order to focus students attention so they will move from: figuring out the answers any way they can to...
More informationNumbers and Operations in Base 10 and Numbers and Operations Fractions
Numbers and Operations in Base 10 and Numbers As the chart below shows, the Numbers & Operations in Base 10 (NBT) domain of the Common Core State Standards for Mathematics (CCSSM) appears in every grade
More informationComparing Fractions and Decimals
Grade 4 Mathematics, Quarter 4, Unit 4.1 Comparing Fractions and Decimals Overview Number of Instructional Days: 10 (1 day = 45 60 minutes) Content to be Learned Explore and reason about how a number representing
More informationRobyn Seifert Decker
Robyn Seifert Decker UltraMathPD@gmail.com Place Value Addition Subtraction Problem Solving Fractions If time allows: Multiplication and Division Spiral of change From Prochaska, DiClemente & Norcross,
More informationWendy Bray, Ph.D National Council of Teachers of Mathematics Annual Meeting April 2013
Wendy Bray, Ph.D Wendy.Bray@ucf.edu National Council of Teachers of Mathematics Annual Meeting April 2013 Tracing Fraction Concept Development Common Core State Standards in Mathematics Grades 1 & 2 Grade
More informationFourth Grade Mathematics
Fourth Grade Mathematics By the end of grade four, students develop quick recall of the basic multiplication facts and related division facts. They develop fluency with efficient procedures for multiplying
More informationMath, Grades 13 TEKS and TAKS Alignment
111.13. Mathematics, Grade 1. 111.14. Mathematics, Grade 2. 111.15. Mathematics, Grade 3. (a) Introduction. (1) Within a wellbalanced mathematics curriculum, the primary focal points are adding and subtracting
More informationSample Fraction Addition and Subtraction Concepts Activities 1 3
Sample Fraction Addition and Subtraction Concepts Activities 1 3 College and CareerReady Standard Addressed: Build fractions from unit fractions by applying and extending previous understandings of operations
More informationPlanning Guide. Number Specific Outcome 3
Mathematics Planning Guide Grade 4 Addition and Subtraction Number Specific Outcome 3 This Planning Guide can be accessed online at: http://www.learnalberta.ca/content/mepg4/html/pg4_additionandsubtraction/index.html
More informationEureka Math Tips for Parents
Eureka Math Tips for Parents Place Value and Decimal Fractions In this first module of, we will extend 4 th grade place value work to multidigit numbers with decimals to the thousandths place. Students
More informationDivision with Whole Numbers and Decimals
Grade 5 Mathematics, Quarter 2, Unit 2.1 Division with Whole Numbers and Decimals Overview Number of Instructional Days: 15 (1 day = 45 60 minutes) Content to be Learned Divide multidigit whole numbers
More informationFraction Vocabulary. It is important that vocabulary terms are taught to students.
Rational Numbers Fractions Decimals Percents It is important for students to know how these 3 concepts relate to each other and how they can be interchanged. Fraction Vocabulary It is important that vocabulary
More informationAbout Singapore Math. Fewer Topics; More Depth. Start with the Concrete. vii
About Singapore MATH About Singapore Math This math course is based on the textbook series Math in Focus, which was developed using the principles of the highly successful Singapore math program. Singapore
More informationRepton Manor Primary School. Maths Targets
Repton Manor Primary School Maths Targets Which target is for my child? Every child at Repton Manor Primary School will have a Maths Target, which they will keep in their Maths Book. The teachers work
More informationFOURTH GRADE NUMBER SENSE
FOURTH GRADE NUMBER SENSE Number sense is a way of thinking about number and quantity that is flexible, intuitive, and very individualistic. It grows as students are exposed to activities that cause them
More informationInvestigations in Number, Data, and Space
Investigations in Number, Data, and Space First Edition Impact Study MODES OF TEACHING AND WAYS OF THINKING Anne Goodrow TERC/Tufts University Abstract From drawing a set or writing numerals to represent
More informationGrade 7 Mathematics. Unit 5. Operations with Fractions. Estimated Time: 24 Hours
Grade 7 Mathematics Operations with Fractions Estimated Time: 24 Hours [C] Communication [CN] Connections [ME] Mental Mathematics and Estimation [PS] Problem Solving [R] Reasoning [T] Technology [V] Visualization
More informationMeasurement with Ratios
Grade 6 Mathematics, Quarter 2, Unit 2.1 Measurement with Ratios Overview Number of instructional days: 15 (1 day = 45 minutes) Content to be learned Use ratio reasoning to solve realworld and mathematical
More informationFIDDLIN WITH FRACTIONS
FIDDLIN WITH FRACTIONS Special Area: Connections (Fourth and Fifth Grade) Written by: (Dawn Ramos, Joan Johnson, Mindy Wilshusen, St. Mary s School) Length of Unit: (6 Lessons) I. ABSTRACT The purpose
More informationPlanning Guide. Number Specific Outcomes 5 and 6
Mathematics Planning Guide Grade 5 Multiplying and Dividing Whole Numbers Number Specific Outcomes 5 and 6 This Planning Guide can be accessed online at: http://www.learnalberta.ca/content/mepg5/html/pg5_multiplyingdividingwholenumbers/index.html
More informationTYPES OF NUMBERS. Example 2. Example 1. Problems. Answers
TYPES OF NUMBERS When two or more integers are multiplied together, each number is a factor of the product. Nonnegative integers that have exactly two factors, namely, one and itself, are called prime
More informationInstructional Resources/Materials: 1. Paper Strips About 8 paper strips measured 1 x Pencil, crayons or markers 3. Ruler
Grade Level/Course: Grade 3 Lesson/Unit Plan Name: Partitioning Wholes into Parts Rationale/Lesson Abstract: This lesson helps builds student s understanding of fractions. Using geometric shapes, students
More informationMultiply Using the Distributive Property
Multiply Using the Distributive Property Common Core Standard: Use place value understanding and properties of operations to perform multidigit arithmetic. 4.NBT.5 Multiply a whole number of up to four
More informationProblem of the Month. Squirreling it Away
The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards: Make sense of problems
More informationSuch As Statements, Kindergarten Grade 8
Such As Statements, Kindergarten Grade 8 This document contains the such as statements that were included in the review committees final recommendations for revisions to the mathematics Texas Essential
More informationPocantico Hills School District Grade 1 Math Curriculum Draft
Pocantico Hills School District Grade 1 Math Curriculum Draft Patterns /Number Sense/Statistics Content Strands: Performance Indicators 1.A.1 Determine and discuss patterns in arithmetic (what comes next
More informationChapter 111. Texas Essential Knowledge and Skills for Mathematics. Subchapter A. Elementary
Elementary 111.A. Chapter 111. Texas Essential Knowledge and Skills for Mathematics Subchapter A. Elementary Statutory Authority: The provisions of this Subchapter A issued under the Texas Education Code,
More informationparent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN GRADE FIVE
TM parent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN GRADE FIVE 5 America s schools are working to provide higher quality instruction than ever before. The way we taught students in the past simply does
More information2013 Texas Education Agency. All Rights Reserved 2013 Introduction to the Revised Mathematics TEKS: Vertical Alignment Chart Kindergarten Algebra I 1
2013 Texas Education Agency. All Rights Reserved 2013 Introduction to the Revised Mathematics TEKS: Vertical Alignment Chart Kindergarten Algebra I 1 The materials are copyrighted (c) and trademarked (tm)
More informationComposing and Decomposing Whole Numbers
Grade 2 Mathematics, Quarter 1, Unit 1.1 Composing and Decomposing Whole Numbers Overview Number of instructional days: 10 (1 day = 45 60 minutes) Content to be learned Demonstrate understanding of mathematical
More informationRational Number Project
Rational Number Project Fraction Operations and Initial Decimal Ideas Lesson : Overview Students estimate sums and differences using mental images of the 0 x 0 grid. Students develop strategies for adding
More informationTeaching & Learning Plans. The Multiplication of Fractions. Junior Certificate Syllabus
Teaching & Learning Plans The Multiplication of Fractions Junior Certificate Syllabus The Teaching & Learning Plans are structured as follows: Aims outline what the lesson, or series of lessons, hopes
More informationNCTM Curriculum Focal Points for Grade 5. Everyday Mathematics, Grade 5
NCTM Curriculum Focal Points and, Grade 5 NCTM Curriculum Focal Points for Grade 5 Number and Operations and Algebra: Developing an understanding of and fluency with division of whole numbers Students
More informationGrade 3 Mathematics Assessment. Eligible Texas Essential Knowledge and Skills
Grade 3 Mathematics Assessment Eligible Texas Essential Knowledge and Skills STAAR Grade 3 Mathematics Assessment Reporting Category 1: Numbers, Operations, and Quantitative Reasoning The student will
More informationKathleen Cramer, Debra Monson, Stephanie Whitney, Seth Leavitt, and Terry Wyberg
Copyright 200 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written
More informationCourse Title: Math Grade Level: Fourth
Course Title: Math Grade Level: Fourth Math  Fourth Page 1 2.1 Numbers, Number Systems and Number Relationships: A. Use expanded notation to represent whole numbers or decimals. B. Apply number theory
More informationFirst Six Weeks (29 Days)
Mathematics Scope & Sequence Grade 6 Revised: May 2010 Topic Title Unit 1: Compare and Order, Estimate, Exponents, and Order of Operations Lesson 11: Comparing and Ordering Whole Numbers Lesson 12: Estimating
More informationAddition Strategies That Use Doubles Facts
Addition Strategies That Use Doubles Facts Objective To provide opportunities for children to explore and practice doublesplus and doublesplus facts. www.everydaymathonline.com epresentations etoolkit
More informationEveryday Mathematics. Grade 4 GradeLevel Goals CCSS EDITION. Content Strand: Number and Numeration. Program Goal Content Thread GradeLevel Goal
Content Strand: Number and Numeration Understand the Meanings, Uses, and Representations of Numbers Understand Equivalent Names for Numbers Understand Common Numerical Relations Place value and notation
More informationFraction Models Grade Three
Ohio Standards Connection Number, Number Sense and Operations Benchmark C Represent commonly used fractions and mixed numbers using words and physical models. Indicator 5 Represent fractions and mixed
More informationProgress Check 6. Objective To assess students progress on mathematical content through the end of Unit 6. Looking Back: Cumulative Assessment
Progress Check 6 Objective To assess students progress on mathematical content through the end of Unit 6. Looking Back: Cumulative Assessment The MidYear Assessment in the Assessment Handbook is a written
More informationPlanning Guide. Grade 6 Factors and Multiples. Number Specific Outcome 3
Mathematics Planning Guide Grade 6 Factors and Multiples Number Specific Outcome 3 This Planning Guide can be accessed online at: http://www.learnalberta.ca/content/mepg6/html/pg6_factorsmultiples/index.html
More informationEveryday Mathematics. Grade 4 GradeLevel Goals. 3rd Edition. Content Strand: Number and Numeration. Program Goal Content Thread GradeLevel Goals
Content Strand: Number and Numeration Understand the Meanings, Uses, and Representations of Numbers Understand Equivalent Names for Numbers Understand Common Numerical Relations Place value and notation
More informationNumberLine Posters for Fractions
NumberLine Posters f Fractions Objective To introduce the number line as a model f fractions. www.everydaymathonline.com epresentations etoolkit Algithms Practice EM Facts Wkshop Game Family Letters Assessment
More informationRational Number Project
Rational Number Project Fraction Operations and Initial Decimal Ideas Lesson 10: Overview Students develop an understanding of thousandths and begin to look at equivalence among tenths, hundredths, and
More informationCommon Core State Standards. Standards for Mathematical Practices Progression through Grade Levels
Standard for Mathematical Practice 1: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for
More informationTexas Assessment of Knowledge and Skills (TAKS) 6th Grade
Texas Assessment of Knowledge and Skills (TAKS) 6th Grade 98 99 100 Grade 6 Mathematics TAKS Objectives and TEKS Student Expectations TAKS Objective 1 The student will demonstrate an understanding of numbers,
More informationInteger Operations. Overview. Grade 7 Mathematics, Quarter 1, Unit 1.1. Number of Instructional Days: 15 (1 day = 45 minutes) Essential Questions
Grade 7 Mathematics, Quarter 1, Unit 1.1 Integer Operations Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Describe situations in which opposites combine to make zero.
More informationCCSS.Math.Content.7.NS.A.3 Solve realworld and mathematical problems involving the four operations with rational numbers. 1
Grade Level/Course: th grade Mathematics Lesson/Unit Plan Name: Complex Fractions Rationale/Lesson Abstract: For the first time, th graders are being asked to work with complex fractions. Timeframe: Introduction
More informationUnpacking Division to Build Teachers Mathematical Knowledge
Unpacking Division to Build Teachers Mathematical Knowledge Melissa Hedges, DeAnn Huinker, and Meghan Steinmeyer University of WisconsinMilwaukee November 2004 Note: This article is based upon work supported
More informationTopic: 1  Understanding Addition and Subtraction
8 days / September Topic: 1  Understanding Addition and Subtraction Represent and solve problems involving addition and subtraction. 2.OA.1. Use addition and subtraction within 100 to solve one and twostep
More informationNumberLine Posters for Fractions
NumberLine Posters f Fractions Objective To introduce the number line as a model f fractions. www.everydaymathonline.com epresentations etoolkit Algithms Practice EM Facts Wkshop Game Family Letters Assessment
More informationMathematical Proficiency By Kenneth Danley Principal, Starline Elementary School1
W e know that parents are eager to help their children be successful in school. Many parents spend countless hours reading to and with their children. This is one of the greatest contributions a parent
More informationBedford Public Schools
Bedford Public Schools Grade 4 Math The fourth grade curriculum builds on and extends the concepts of number and operations, measurement, data and geometry begun in earlier grades. In the area of number
More informationCindy s Cats. Cindy has 3 cats: Sammy, Tommy and Suzi.
Cindy s Cats This problem gives you the chance to: solve fraction problems in a practical context Cindy has 3 cats: Sammy, Tommy and Suzi. 1. Cindy feeds them on Cat Crunchies. Each day Sammy eats 1 2
More informationUnit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.
Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L34) is a summary BLM for the material
More information1 Teaching the Lesson
Getting Started Mental Math and Reflexes Children solve division number stories. They write answers on slates and explain their strategies to the class. Provide counters. There are children in the class.
More informationPlanning Guide. Number Specific Outcomes 8, 9, 10 and 11
Mathematics Planning Guide Grade 5 Working with Decimal Numbers Number Specific Outcomes 8, 9, 10 and 11 This Planning Guide can be accessed online at: http://www.learnalberta.ca/content/mepg5/html/pg5_workingwithdecimalnumbers/index.html
More informationOverview. Essential Questions. Grade 5 Mathematics, Quarter 3, Unit 3.1 Adding and Subtracting Decimals
Adding and Subtracting Decimals Overview Number of instruction days: 12 14 (1 day = 90 minutes) Content to Be Learned Add and subtract decimals to the hundredths. Use concrete models, drawings, and strategies
More informationLevel Descriptors Maths Level 15
Level Descriptors Maths Level 15 What is APP? Student Attainment Level Descriptors APP means Assessing Pupil Progress. What are the APP sheets? We assess the children in Reading, Writing, Speaking & Listening,
More informationPrentice Hall Mathematics: Course Correlated to: Alaska State Content Standards: Math (Grade 7)
Alaska State Content Standards: Math (Grade 7) A. A student should understand mathematical facts, concepts, principles, and theories. 1. understand and use numeration, including numbers, number systems,
More informationThe Crescent Primary School Calculation Policy
The Crescent Primary School Calculation Policy Examples of calculation methods for each year group and the progression between each method. January 2015 Our Calculation Policy This calculation policy has
More informationPlanning Guide. Number Specific Outcomes 8, 9, 10
Mathematics Planning Guide Grade Fractions and Decimals Number Specific Outcomes 8, 9, 0 This Planning Guide can be accessed online at: http://www.learnalberta.ca/content/mepg/html/pg_fractionsanddecimals/index.html
More informationPerformance Assessment Task Bikes and Trikes Grade 4. Common Core State Standards Math  Content Standards
Performance Assessment Task Bikes and Trikes Grade 4 The task challenges a student to demonstrate understanding of concepts involved in multiplication. A student must make sense of equal sized groups of
More informationMathematical goals. Starting points. Materials required. Time needed
Level N of challenge: B N Mathematical goals Starting points Materials required Time needed Ordering fractions and decimals To help learners to: interpret decimals and fractions using scales and areas;
More informationUpon successful completion of this course you should be able to
University of Massachusetts, Amherst Math 113 Math for Elementary School Teachers I Fall 2011 Syllabus of Objectives and Learning Outcomes Overview of the Course This is a mathematics content course which
More informationModule 2: Working with Fractions and Mixed Numbers. 2.1 Review of Fractions. 1. Understand Fractions on a Number Line
Module : Working with Fractions and Mixed Numbers.1 Review of Fractions 1. Understand Fractions on a Number Line Fractions are used to represent quantities between the whole numbers on a number line. A
More informationDanville District No. 118 Mathematics Fifth Grade Curriculum and Scope and Sequence First Quarter
Danville District No. 118 Mathematics Fifth Grade Curriculum and Scope and Sequence First Quarter Common Core Operations and Algebraic Thinking (5.OA) Common Core Number and Operations in Base Ten (5.NBT)
More informationRational Number Project
Rational Number Project Fraction Operations and Initial Decimal Ideas Lesson 1: Overview Students review how to model fractions with fraction circles by ordering unit fractions, using 1half as a benchmark
More information3rd Grade Lesson Fractions
3rd Grade Lesson Fractions Lesson planning team: Tracey Carter, Rebecca Kotler, Tom McDougal Instructor: Tracey Carter Focus Question Sixth Annual Lesson Study Conference DePaul University Chicago, Illinois
More informationAERO/Common Core Alignment 35
AERO/Common Core Alignment 35 Note: In yellow are the AERO Standards and inconsistencies between AERO and Common Core are noted by the strikethrough ( eeeeee) notation. AERO Common Core Mapping 1 Table
More informationFRACTIONS: EASY AS PIE! Grade Level: 4th Grade Presented by: Sue Gill, Clegern Elementary School, Edmond, Oklahoma Length of Unit: 7 lessons
FRACTIONS: EASY AS PIE! Grade Level: 4th Grade Presented by: Sue Gill, Clegern Elementary School, Edmond, Oklahoma Length of Unit: 7 lessons I. ABSTRACT This unit, "Fractions: Easy as Pie! is intended
More informationHigh School Functions Building Functions Build a function that models a relationship between two quantities.
Performance Assessment Task Coffee Grade 10 This task challenges a student to represent a context by constructing two equations from a table. A student must be able to solve two equations with two unknowns
More informationNew Ontario Mathematics Curriculum (K 6) Correlation to the Number Developmental Map
Correlation to the Number Developmental Map to 10 to 100 Modelled ly to Concept 1 Numbers tell how many or how much. K: investigate the idea that quantity is greater when counting forwards and less when
More informationEnhanced Instructional Transition Guide
Addition and Subtraction with Whole Numbers and Decimals (13 days) Possible Lesson 01 (5 days) Possible Lesson 02 (8 days) POSSIBLE LESSON 01 (5 days) This lesson is one approach to teaching the State
More informationRATIONAL NUMBER ADDITION AND SUBTRACTION
21 RATIONAL NUMBER ADDITION AND SUBTRACTION INSTRUCTIONAL ACTIVITY Lesson 4 LEARNING GOAL Students will extend their understanding of integer addition and subtraction to addition and subtraction of rational
More informationCommon Core Mathematics Challenge
Level: Domain: Cluster: Grade Five Number and Operations Fractions Use equivalent fractions as a strategy to add and subtract fractions. Standard Add and subtract fraction with unlike denominators (including
More information