Multidigit Multiplication: Teaching methods and student mistakes


 Sharlene Simpson
 2 years ago
 Views:
Transcription
1 Cynthia Ashton Math Professor Beck Term Paper Due: Friday 14 December 2012 Multidigit Multiplication: Teaching methods and student mistakes Abstract: Description of the paper. Why is it important/interesting, what are your findings? In this paper I will address the importance of conceptual understanding of mathematics regarding algorithm of multidigit multiplication. Often times in mathematics, students tend to believe the purpose of solving a mathematical equation is to find the answer, through the procedure of the algorithm. In solving a mathematical equation, it is more valuable for the intellectual development of the students by understanding algorithms conceptually. Through my research I discovered the importance of algorithmic procedure in which is not limited in one method, but can have many different algorithms to solve the same equation. Within my research I've also found the importance of mental algorithms rather than written formal algorithm tactics, in regards to the impacts of a student s ideology of mathematical proofs. Students have a difficult time of conceptually understanding the explanations of multidigit multiplication. Regarding to the book Knowing and Teaching Elementary Mathematics by Liping Ma, the issues regarding multidigit multiplication is that "students often forget 'to move the numbers' over each line" (Ma, 28). Ma is referring to the "caring out the liningup procedure algorithm, which most Americans were taught with, but don t conceptually
2 understand the reason for this method is because of place value for the digit. Ma also points out that the U.S. teaching methods vary from procedural methods to conceptual explanations, where she compares Chinese teaching methods (Ma, 51). Within this term paper I will address the importance of conceptual teaching methods regarding multidigit multiplication. I will also touch bases of the significance of algorithms tied to Ma s thematic proofs that relate to multiplication. This topic was interesting to me because it explained why students in the U.S. struggle with math, it is because students lack of conceptual understanding because student learn algorithmic procedure to solve equations. Introduction: Issues/problem, significance, solution Recently when I was doing my multidigit multiplication equation for my Math Concept course, I was questioning if 200 x 400 is 8000 or 80,000. Students often have an issue of understanding the concept of place value while solving a multidigit multiplication equation. Because of this, students place numbers within the wrong column or wrong location of the place value that leads to the incorrect answer. Liping Ma addresses the issue from the styles of teaching methods between teachers of the United States and China. Teachers in the United States practice more on a procedural teaching method compared to the teachers in China practice conceptual teaching methods. These two types of methods are significant to a student s understanding in algorithms, mental computation and mathematical proofs. If students don't have a basic comprehension of the basic mathematics knowledge, students will struggle with higher levels of mathematics. Conceptual teaching is the solution for successful multidigit multiplication, which will lead to further success for the student s mathematical career
3 and daily math problems within life itself. Method: How you went about searching info or data; impression of quality I used Laping Ma's book as a base for my term paper. Constantly referring to her terms and topics within my search on San Francisco State University's library catalog and articles & databases. Many references went into too much detail of the analysis of topics like mathematical proofs, algorithms, multidigit multiplication, and place value. My main focus is on conceptual teaching where many articles did not reflect the subject matter of mathematics. I've decided to use the support of mathematical proofs and algorithms to further rationalize the importance of conceptual learning. Paragraph #1 The Issue Teaching multidigit multiplication by procedural methods does not give the student proper understanding of place value and the distributive law. 70% of United States teachers said the problem was incorrect liningup procedure, whereas the 30% concluded that the students did not understand the rational of the algorithm (Ma, 29). In the school of mathematics, an algorithm is a systematic method used for solving an equation. By following the procedure one can solve a mathematical equation that gives an output correctly or announce that it cannot be solved (Morrow, 21). In the school of mathematics algorithms gives procedure to solve an equation, yet the equation can have many different kinds of algorithm that will result in the same output. Within the issue of multidigit multiplication Ma focuses on the algorithm of "caring out the liningup procedure"(ma, 29). In trying to calculate:
4 27 X 24 Students seem to be forgetting to "move the numbers" (I.e. the partial products) over on each line (Ma, 28). Students carried out the equation like this: Instead of this: 27 X X Within Ma's study all the teachers agreed that it is a "problem of mathematical learning rather than a careless oversight" (Ma, 29). All teachers agreed that there was a problem with the learning comprehension for the students, which is a direct refection of the teaching methods of the teacher. Even though teachers have difficulty teaching multidigit multiplication and notices similar mistakes every year when it is taught, teachers don't seem to take much action in changing teaching methods in order to provide better comprehension of the subject matter for the students. The carrying out the lining up algorithm is taught with a procedurally directed method, refers the term "place value" as the location of the numbers (Ma, 29). The procedurally directed approach "verbalized the algorithm so it can be carried out correctly" yet by doing these teachers are not providing the understanding of the importance of the definition of true place value (Ma, 29).
5 Although teachers used other methods like using lined paper or a grid to position the zeros in the placeholder, teacher merely suggested to place the numbers correctly (Ma, 34). The term place value was not introduced to students as a mathematical concept, but as labels for columns where they should put numbers (Ma, 34). By putting artificial value into the meaning of place holder students lack an understanding of place value. With multidigit multiplication students need to comprehend the mathematical terms in order to solve with understanding of the algorithm(s). By defining these mathematical terms students will have a conceptual understanding of these terms that will help their future academic success with mathematics. Paragraph 2: Algorithms and Multidigit Multiplication In mathematics, algorithms are executed to announce the correct answer for the equation. For multidigit multiplication I found three different formal written algorithms besides the "caring out the liningup procedure" method that Ma heavily focused within her book. Nonetheless within the classroom students learn the formal written algorithmic procedures, yet mental computation strategies are just as valuable. According to Marrow, mental computation is solving mathematics equations, which are solved by a student s personal choice of strategy that is based on the numbers given (44). One can say it is the mathematical solving method that a student solves "in their head. This gives flexibility along with the required understanding of algorithm. Marrow further states the value of mental computation: Those children have powers of mental computation that
6 they develop themselves with out support. The heart of all these selfdevised algorithms is that the child tries to turn a difficult calculation into an easy one: is hard, is easy; 9x17 is hard, 10x1717 is easier. The more able children develop their ability to devise algorithms and so make life easy. Unfortunately the less able children are less capable of helping themselves and so are left with much more difficult task (46). Mentalarithmetic procedure allows students to realize the different ways of solving an equation mentally instead of limiting students to one algorithmic procedure. Formal written algorithms are helpful for visual learners. Yet others argue "teaching formal algorithms can be counter productive" (Marrow, 44). Regarding multidigit multiplication Marrow introduces four algorithms. I will go into details with three of them. First formal written algorithm Marrow introduces is the direct modeling strategy, which is the same as Professor Beck introduced to his Math Concept course (Marrow, 152; Notes, 10/10/2012). The direct modeling, sorts the by tens and ones to create simplicity for the multidigit multiplication equation as long as making it visual for the students (Marrow, 153). For example 28X12 is modeled out like this: will be place in the digit s proper place value outside the boxes, 20 & 8 or 2 tens and 8 ones. 12 will also be placed in their proper place value outside the boxes which will be 10 & 2 or 1 tens and 2 ones. Then one will proceed to solve the equation by multiplying 20x10= 200 and will fill it in within the box. The steps will apply to fill each box like this:
7 Finally one will use addition to add all the results like this: =336 Given to final answer to the equation, which are 336. By using the direct modeling method student can practice breaking up the number by place value, which helps them conceptually understand place value. This also makes it easier to solve the equation because the numbers are whole or end in a 0. The second formal written algorithm that Marrow introduces is the complete number strategy. The complete number strategies are based on repeated addition, which is an adding procedure by doubling the multiplicands (Marrows, 152).
8 For example the equation 28x12 will be solved like this: Complete number strategies are repeated addition or doubling (Marrow, 153). This method shows the student exactly what multiplication does, which is repeated addition. By doing this method students can visually see that 28X12 is = 336. The third algorithm Marrow introduced is the partitioning number strategy, which is spit the multiplicand or multiplier into two or more numbers and create multiple subproblems that are easier to deal with (152).
9 For example the equation 177x3 will be solved like this: This allows students to make the digits easier by grouping them into simpler numbers. All of the three of Marrow's algorithms did not mention within Ma's issue regarding the "caring out the liningup procedure" algorithm. Along with Marrow not mentioning
10 anything regarding place value and the distributive property. Perhaps it is because he too is procedurally explaining the rational of the algorithm and not emphasizing on the conceptual teaching. It is important that students understand the concepts by being able to identify the terms of mathematics. Regardless to say that it is important for student to learn the different ways of solving the problem in order to find our the student's preferred method by experiences more than one algorithm. Marrow does introduce the importance of mental computation, which is branch to the importance of mathematical proofs. Paragraph 3: Extent of the Problem: Mathematical proofs and importance of Multiplication Mathematical proof has a cognitive and socialcultural impact upon the student s development. Referring to the article Toward comprehensive perspectives on the learning and teaching of proof, the article refers to the historicalepistemological, cognitive, and instructionalsocialcultural of mathematical proof in which directly reflects and relates to the importance of multiplication. According to the article, proofs establish the truth for a person and/or a community (Harel, 3). Mathematical algorithms give precise systematic methods that announce an output. These algorithms announce mathematical proofs. The concept of proof consecutively gives an individual assertion that who is uncertain of truth that then becomes a fact that the individual is certain of the truth (Harel, 6). The ideology of proofs, justifies the importance of mathematics and how it applies to our daily lives. This proving process removes doubts of truth of the assertion (Harel, 6). This sense making builds one's confidence in assertion but also influences others by leading to a social practice of persuading (Harel, 8). Proof gives students truths, and with
11 multiplication students can continue to exercise these truths throughout their academic career and daily life. Morrow further states: Understanding multiplication is central to knowing mathematics. The curriculum and evaluation standards for School Mathematics (National Council of Teachers of Mathematics 1989) proposed that children need to develop meaning for multiplication by creating algorithms and procedures for operations (Morrow, 151). Multiplication is considered to be the core of mathematics. According to Jerama, the times tables is considered to be the building blocks for the higher schools of mathematics (Jarema). Mental computation is directly connected to proofs, which is used in our daily lives unconsciously because they are not traditional written algorithms often used within a classroom setting. When solving a multidigit multiplication problem students need to understand the importance of place value in order to carry the algorithmic procedure correctly this reflects the concept of proof which is also linked to mental computation. The teaching method of perceiving these algorithms impacts the student s knowledge of multiplication. Paragraph 4: Conceptual vs. Procedural i.e. solution A student should learn the mathematical algorithm that justifies and explains the output of the equation. Ma compared U.S. teaching methods to China's. Teachers in China introduced multidigit multiplication first as distributive property as the process of transformation (Ma, 40).
12 First the teacher will compose the problem into three smaller problems like this: 127x249= 127x( ) = 127x x40+127x9 = = Then the teacher will state that the transformation accomplish by teh distributive law (Ma, 40). With the power of distributive law, 249 is broken down to 2 hundreds, 4 tens, and 9 ones. By doing this teachers are emphasizing the importance of place value. Next the teacher will rewrite the equation into columns: 127 x The three partial products are represented in the form of columns (Ma, 41). The teacher then ask students to observe the zeros then erases the zeros which then the equation will appear to be a staircase like this: 127 x Referring to Ma, the features of a mathematical argument justification, rigorous
13 reasoning, and correct expression were reflected throughout [her] explanation (41). The bridge of connection presents conceptual methods of teachers from China makes the algorithm into a true proof which makes it easier for the student to absorb and comprehend what multidigit multiplication does. Going into further explanation, the teachers justify why the zeros can be erased through thorough explanation of place value. When using distributive law, this allows breaking down the numbers. Marrow stated earlier that multiplication is repeated addition. With multidigit multiplication, it is also repeated addition but on a larger scale in which place values identifies the size of the number. For example the number 325, the 5 represents the ones place, the 2 represents the tens place, and the 3 represents the hundredths place. Also commonly can be written out as: Or verbally can be read as: 3 hundredths, 20tenths, and 5 ones. 300 can also be read at 30 tens or 300 ones. By understanding how to deal with several tens, students will be able to grasp the concept of what digit at a certain place stands for (Ma, 4445). Students cannot get a thorough understanding of place value in one day (Ma, 45). Like any subject matter students need to be reminded of the concepts and teacher should observe student s mistakes and repeat explanation of the concepts as needed. By exposing students to explanations of what the zeros represent, students have a broader mathematical perspectives as long with developing their capacity to make a mathematical judgment (Ma, 45).
14 Discussion: What does the observations mean, summarize important findings Teachers who did not conceptual understand the meaning behind the algorithmic procedure lack understanding of the place value and distributive property, which causes to teach the procedural method that can constrain the student s growth of knowledge in math. Although the teachers know how to solve the problem teachers lacking a simple explanation of place value means students will also lack of math concepts. In my experiences of teaching English as a foreign language I often had a difficult time explaining why English grammars behaves in a certain way. When students ask me why does putting an e on a word make a long vowel I have no true explanation because I lack knowledge of the conceptual reasoning of why that the English language behaves that way. It is difficult for an adult to simplify our knowledge because knowledge is constantly being built upon, which leads an adult to easily forget the basic elements of the fundamentals. Although an individual solves the equation correctly, it doesn't necessarily mean the individual's development in mathematics is progressing. Within my research I found the importance of learning many ways of solving the problem through the use of algorithms. By knowing multiply algorithms to solve an equation teachers can use the support of the different procedure to prove the rationalization of the equation. The mental computation is equally as important as the formal written algorithms in order to apply problem solving by second nature. Mathematics is important for the cognitive and socialcultural aspect of the student's ideology of proof that connects to every day life. By mastering and memorizing multiplication students will have a solid foundation for their development of mathematics along with solving equations within everyday life. For
15 example, Susie needs 300 crackers for her block of cheese. She can buy a pack of crackers of 3 sets of 12 or a block of crackers of 5 sets of 10. Conclusion: Regarding to Liping Ma's survey more teachers teach with a procedural directed teaching method rather then a conceptual explanation of rational teaching method in the U.S. (Ma, 53). Teachers can use these two different algorithmic procedures to support the proof of one another. Teaching on an elementary level heavily depends on the teacher's knowledge of the material. Rather than being able to solve a problem just by memorizing the procedures it is more valuable to the student to memorizing the algorithm based on conceptual understanding and the algorithmic proofs of reasoning. Teachers need to approach student mistakes as an opportunity to engage students to identify characteristics of the equation and understanding mathematical concepts. Although the U.S. teachers teach conceptual material it is directed in a procedural method that lack thorough understanding for both the teacher and student. Compared to the Chinese conceptual understanding with a conceptual directed teaching, the mistake students make are identified and attached together.
16 Works Cited Ma, Liping. Knowing and Teaching Elementary Mathematics. Anniversary Edition. New York: Routledge, Print. Morrow, Lorna, and Margaret Kenney. The Teaching and Learning of Algorithms in School Mathematics. Reston: The National Council of Teachers of Mathematics, Inc., ,4448, Print. G. Harel & L. Sowder, Toward comprehensive perspectives on the learning and teaching of proof; G. Hanna, Proofs that prove and proofs that explain. In: G. Vergnaud, J. Rogalski, and M. Artigue (Eds.), Proceedings of the 13th Meeting of the International Group for the Psychology of Mathematics Education (pp ), PME, 1989 Jerema, Susuna. "The Importance of Memorizing Times Tables." Phamtom Writers. ( ): n. page. Web. 28 Oct <
Decimals 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 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 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 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 informationWhat Is Singapore Math?
What Is Singapore Math? You may be wondering what Singapore Math is all about, and with good reason. This is a totally new kind of math for you and your child. What you may not know is that Singapore has
More informationAn Investigation into Visualization and Verbalization Learning Preferences in the Online Environment
An Investigation into Visualization and Verbalization Learning Preferences in the Online Environment Dr. David Seiler, Assistant Professor, Department of Adult and Career Education, Valdosta State University,
More informationThree approaches to oneplace addition and subtraction: Counting strategies, memorized facts, and thinking tools. Liping Ma
Three approaches to oneplace addition and subtraction: Counting strategies, memorized facts, and thinking tools Liping Ma In many countries, the first significant chunk of elementary mathematics is the
More information#1 Make sense of problems and persevere in solving them.
#1 Make sense of problems and persevere in solving them. 1. Make sense of problems and persevere in solving them. Interpret and make meaning of the problem looking for starting points. Analyze what is
More informationOverall Frequency Distribution by Total Score
Overall Frequency Distribution by Total Score Grade 8 Mean=17.23; S.D.=8.73 500 400 Frequency 300 200 100 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
More informationAugust 7, 2013 Elementary Math Summit
August 7, 2013 Elementary Math Summit Beth Edmonds Math Intervention Teacher Wake County Public School System eedmonds@wcpss.net Participate actively. Have an openmind. To become more familiar with Common
More informationTeaching to the Common Core mathematics standards. Why?
Comments? Like PDK at www. facebook.com/pdkintl Why the Common Core changes math instruction It s not the New Math exactly, but the Common Core calls for sharp changes in how math is taught and ultimately
More informationUnderstanding Place Value of Whole Numbers and Decimals Including Rounding
Grade 5 Mathematics, Quarter 1, Unit 1.1 Understanding Place Value of Whole Numbers and Decimals Including Rounding Overview Number of instructional days: 14 (1 day = 45 60 minutes) Content to be learned
More informationA Study in Learning Styles of Construction Management Students. Amit Bandyopadhyay, Ph.D., PE, F.ASCE State University of New York FSC
A Study in Learning Styles of Construction Management Students Amit Bandyopadhyay, Ph.D., PE, F.ASCE State University of New York FSC Abstract Students take in and process information in different ways.
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 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 informationPA Common Core Standards Standards for Mathematical Practice Grade Level Emphasis*
Habits of Mind of a Productive Thinker Make sense of problems and persevere in solving them. Attend to precision. PA Common Core Standards The Pennsylvania Common Core Standards cannot be viewed and addressed
More informationAn Introduction to Various Multiplication Strategies
An Introduction to Various Multiplication Strategies Lynn West Bellevue, NE In partial fulfillment of the requirements for the Master of Arts in Teaching with a Specialization in the Teaching of Middle
More informationUnderstanding Place Value
Grade 5 Mathematics, Quarter 1, Unit 1.1 Understanding Place Value Overview Number of instructional days: 7 (1 day = 45 minutes) Content to be learned Explain that a digit represents a different value
More informationOperations and Algebraic Thinking Represent and solve problems involving multiplication and division.
Performance Assessment Task The Answer is 36 Grade 3 The task challenges a student to use knowledge of operations and their inverses to complete number sentences that equal a given quantity. A student
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 informationElementary School Mathematics Priorities
Elementary School Mathematics Priorities By W. Stephen Wilson Professor of Mathematics Johns Hopkins University and Former Senior Advisor for Mathematics Office of Elementary and Secondary Education U.S.
More informationTHE IMPACT OF DEVELOPING TEACHER CONCEPTUAL KNOWLEDGE ON STUDENTS KNOWLEDGE OF DIVISION
THE IMPACT OF DEVELOPING TEACHER CONCEPTUAL KNOWLEDGE ON STUDENTS KNOWLEDGE OF DIVISION Janeen Lamb and George Booker Griffith University, Brisbane, Australia This study investigated children s knowledge
More informationWriting a Course Paper. Capella University 225 South 6th Street, 9th Floor Minneapolis, MN 55402 1888CAPELLA (2273552)
Writing a Course Paper Capella University 225 South 6th Street, 9th Floor Minneapolis, MN 55402 1888CAPELLA (2273552) Table of Contents Creating Major Sections... 3 Writing Fundamentals... 7 Expressing
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 information+ Addition + Tips for mental / oral session 1 Early addition. Combining groups of objects to find the total. Then adding on to a set, one by one
+ Addition + We encourage children to use mental methods whenever possible Children may not need to be taught every step STEP Tips for mental / oral session Counting Concept & images Comments Early addition
More informationPolynomials and Factoring. Unit Lesson Plan
Polynomials and Factoring Unit Lesson Plan By: David Harris University of North Carolina Chapel Hill Math 410 Dr. Thomas, M D. 2 Abstract This paper will discuss, and give, lesson plans for all the topics
More informationTIPS FOR WRITING LEARNING OBJECTIVES
TIPS FOR WRITING LEARNING OBJECTIVES N ational ACEP receives numerous requests from chapters for assistance on how to write good learning objectives. The information presented in this section has been
More information7 critical reading strategies and activities to do with students to encourage and develop critical reading ability
7 critical reading strategies and activities to do with students to encourage and develop critical reading ability This text has been adapted from and extends on a text written by Salisbury University
More informationAbstraction in Computer Science & Software Engineering: A Pedagogical Perspective
Orit Hazzan's Column Abstraction in Computer Science & Software Engineering: A Pedagogical Perspective This column is coauthored with Jeff Kramer, Department of Computing, Imperial College, London ABSTRACT
More informationSFUSD Mathematics Core Curriculum Development Project
1 SFUSD Mathematics Core Curriculum Development Project 2014 2015 Creating meaningful transformation in mathematics education Developing learners who are independent, assertive constructors of their own
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 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 informationPROSPECTIVE MIDDLE SCHOOL TEACHERS KNOWLEDGE IN MATHEMATICS AND PEDAGOGY FOR TEACHING  THE CASE OF FRACTION DIVISION
PROSPECTIVE MIDDLE SCHOOL TEACHERS KNOWLEDGE IN MATHEMATICS AND PEDAGOGY FOR TEACHING  THE CASE OF FRACTION DIVISION Yeping Li and Dennie Smith Texas A&M University, U.S.A. In this paper, we investigated
More informationThe Mathematics School Teachers Should Know
The Mathematics School Teachers Should Know Lisbon, Portugal January 29, 2010 H. Wu *I am grateful to Alexandra AlvesRodrigues for her many contributions that helped shape this document. Do school math
More informationMULTIPLICATION. Present practical problem solving activities involving counting equal sets or groups, as above.
MULTIPLICATION Stage 1 Multiply with concrete objects, arrays and pictorial representations How many legs will 3 teddies have? 2 + 2 + 2 = 6 There are 3 sweets in one bag. How many sweets are in 5 bags
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 information5Grade. Interactive Notebooks. Ideal for organizing information and applying learning. Perfect for addressing the needs of individual learners
CD1046 Interactive Notebooks 5Grade Ideal for organizing information and applying learning Perfect for addressing the needs of individual learners Includes stepbystep instructions for each page Great
More informationExecutive Summary Principles and Standards for School Mathematics
Executive Summary Principles and Standards for School Mathematics Overview We live in a time of extraordinary and accelerating change. New knowledge, tools, and ways of doing and communicating mathematics
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 informationHerts for Learning Primary Maths Team model written calculations policy Rationale The importance of mental mathematics
Herts for Learning Primary Maths Team model written calculations policy Rationale This policy outlines a model progression through written strategies for addition, subtraction, multiplication and division
More informationGeorgia Standards of Excellence Curriculum Frameworks Mathematics
Georgia Standards of Excellence Curriculum Frameworks Mathematics GSE Coordinate Algebra Unit 2: Reasoning with Equations and Inequalities Unit 2 Reasoning with Equations and Inequalities Table of Contents
More informationLike Terms: What s in a Name?
University of Rhode Island DigitalCommons@URI School of Education Faculty Publications School of Education 2015 Like Terms: What s in a Name? Cornelis de Groot University of Rhode Island, degrootc@uri.edu
More informationDeveloping Base Ten Understanding: Working with Tens, The Difference Between Numbers, Doubling, Tripling, Splitting, Sharing & Scaling Up
Developing Base Ten Understanding: Working with Tens, The Difference Between Numbers, Doubling, Tripling, Splitting, Sharing & Scaling Up James Brickwedde Project for Elementary Mathematics jbrickwedde@ties2.net
More informationNew York State Testing Program Grade 5 Common Core Mathematics Test. Released Questions with Annotations
New York State Testing Program Grade 5 Common Core Mathematics Test Released Questions with Annotations August 2013 THE STATE EDUCATION DEPARTMENT / THE UNIVERSITY OF THE STATE OF NEW YORK / ALBANY, NY
More informationWSMC High School Competition The Pros and Cons Credit Cards Project Problem 2007
WSMC High School Competition The Pros and Cons Credit Cards Project Problem 2007 Soon, all of you will be able to have credit cards and accounts. They are an important element in the fabric of the United
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 informationSupport Booklet for. Student Name: Coach Name(s): Start Date: What does the student want to achieve from this programme?
Support Booklet for Student Name: Name(s): Start : What does the student want to achieve from this programme? Target End : Sarah Emson Page 1 Why and Who? Using Turn Tables Using Turn Tables Individual
More informationName Period Date MATHLINKS: GRADE 7 STUDENT PACKET 1 FRACTIONS AND DECIMALS
Name Period Date 7 STUDENT PACKET MATHLINKS: GRADE 7 STUDENT PACKET FRACTIONS AND DECIMALS. Terminating Decimals Convert between fractions and terminating decimals. Compute with simple fractions.. Repeating
More informationThis lesson introduces students to decimals.
NATIONAL MATH + SCIENCE INITIATIVE Elementary Math Introduction to Decimals LEVEL Grade Five OBJECTIVES Students will compare fractions to decimals. explore and build decimal models. MATERIALS AND RESOURCES
More informationMath Counts: Issues That Matter
Math Counts: Issues That Matter A P R O F E S S I O N A L S E R I E S, V O L U M E 6 MASTERING THE BASIC FACTS nderstanding and learning basic U facts to automaticity is a primary cornerstone in mathematics
More informationSt. Joseph s Catholic Primary School. Calculation Policy April 2016
St. Joseph s Catholic Primary School Calculation Policy April 2016 Information The following calculation policy has been devised to meet requirements of the National Curriculum 2014 for the teaching and
More information0.75 75% ! 3 40% 0.65 65% Percent Cards. This problem gives you the chance to: relate fractions, decimals and percents
Percent Cards This problem gives you the chance to: relate fractions, decimals and percents Mrs. Lopez makes sets of cards for her math class. All the cards in a set have the same value. Set A 3 4 0.75
More informationGRADE 6 MATH: SHARE MY CANDY
GRADE 6 MATH: SHARE MY CANDY UNIT OVERVIEW The length of this unit is approximately 23 weeks. Students will develop an understanding of dividing fractions by fractions by building upon the conceptual
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 information**Unedited Draft** Arithmetic Revisited Lesson 5: Decimal Fractions or Place Value Extended Part 2: Adding and Subtracting Decimals
1. Adding Decimals **Unedited Draft** Arithmetic Revisited Lesson 5: Decimal Fractions or Place Value Extended Part 2: Adding and Subtracting Decimals Decimal arithmetic is very similar to whole number
More informationSum of Rational and Irrational Is Irrational
About Illustrations: Illustrations of the Standards for Mathematical Practice (SMP) consist of several pieces, including a mathematics task, student dialogue, mathematical overview, teacher reflection
More information1. I have 4 sides. My opposite sides are equal. I have 4 right angles. Which shape am I?
Which Shape? This problem gives you the chance to: identify and describe shapes use clues to solve riddles Use shapes A, B, or C to solve the riddles. A B C 1. I have 4 sides. My opposite sides are equal.
More informationMDI01 EXAM PREPARATION & ADVICE
MDI01 EXAM PREPARATION & ADVICE DIPLOMA 1. EXAM STRUCTURE AND MARKING SCHEME The MDI01 exam is 3 hours in duration and has a total of 200 marks available and carry a pass mark of 55% (110 marks to pass).
More informationDECIMALS are special fractions whose denominators are powers of 10.
DECIMALS DECIMALS are special fractions whose denominators are powers of 10. Since decimals are special fractions, then all the rules we have already learned for fractions should work for decimals. The
More informationAcademic Integrity. Writing the Research Paper
Academic Integrity Writing the Research Paper A C A D E M I C I N T E G R I T Y W R I T I N G T H E R E S E A R C H P A P E R Academic Integrity is an impressivesounding phrase. What does it mean? While
More informationCircuits and Boolean Expressions
Circuits and Boolean Expressions Provided by TryEngineering  Lesson Focus Boolean logic is essential to understanding computer architecture. It is also useful in program construction and Artificial Intelligence.
More informationToothpicks and Transformations
The High School Math Project Focus on Algebra Toothpicks and Transformations (Quadratic Functions) Objective Students will investigate quadratic functions using geometric toothpick designs. Overview of
More information( ) 4, how many factors of 3 5
Exponents and Division LAUNCH (9 MIN) Before Why would you want more than one way to express the same value? During Should you begin by multiplying the factors in the numerator and the factors in the denominator?
More informationCCSS.Math.Content.5.OA.A.1 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
Directions: Respond to the prompts below (no more than 8 singlespaced pages, including prompts) by typing your responses within the brackets following each prompt. Do not delete or alter the prompts; both
More informationGRADE 8 MATH: TALK AND TEXT PLANS
GRADE 8 MATH: TALK AND TEXT PLANS UNIT OVERVIEW This packet contains a curriculumembedded Common Core standards aligned task and instructional supports. The task is embedded in a three week unit on systems
More informationISBN:
OnlineResources This work is protected by United States copyright laws and is provided solely for the use of teachers and administrators in teaching courses and assessing student learning in their classes
More informationChapter 11 Number Theory
Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications
More informationSection 1.5 Arithmetic in Other Bases
Section Arithmetic in Other Bases Arithmetic in Other Bases The operations of addition, subtraction, multiplication and division are defined for counting numbers independent of the system of numeration
More informationMathematics at PS 29. Presented by Molly Dubow, Hana Pardon and Kim Van Duzer
Mathematics at PS 29 Presented by Molly Dubow, Hana Pardon and Kim Van Duzer Agenda Introductions Try Some Math: Convincing a Skeptic Our Beliefs about Teaching Math at PS 29 Our Curriculum and Teaching
More informationWhat Is a Classroom Number Talk?
CHAPTER 1 What Is a Classroom Number Talk? Rationale for Recently, during a visit to a secondgrade classroom, I watched Melanie subtract 7 from 13. She had written the problem vertically on her paper
More informationGrade 5 Math Content 1
Grade 5 Math Content 1 Number and Operations: Whole Numbers Multiplication and Division In Grade 5, students consolidate their understanding of the computational strategies they use for multiplication.
More informationA REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
Performance Assessment Task Magic Squares Grade 9 The task challenges a student to demonstrate understanding of the concepts representing and analyzing mathematical situations and structures with algebraic
More informationUnderstanding Primary Children s Thinking and Misconceptions
Presentation Title Understanding Primary Children s Thinking and Misconceptions in Decimal Numbers Format Paper Session [ 8.01 ] Subtheme Assessment and Student Achievement Understanding Primary Children
More informationNew York State Testing Program Grade 3 Common Core Mathematics Test. Released Questions with Annotations
New York State Testing Program Grade 3 Common Core Mathematics Test Released Questions with Annotations August 2013 THE STATE EDUCATION DEPARTMENT / THE UNIVERSITY OF THE STATE OF NEW YORK / ALBANY, NY
More informationXU Xiaofei. Nanhu College, Jiaxing University, Jiaxing, China
USChina Foreign Language, December 2015, Vol. 13, No. 12, 859863 doi:10.17265/15398080/2015.12.002 D DAVID PUBLISHING Genre Approach to English Writing in Chinese College Teaching XU Xiaofei Nanhu
More informationMath: Study Skills, Note Taking Skills, And Test Taking Strategies
Math: Study Skills, Note Taking Skills, And Test Taking Strategies Math Study Skill Active Study vs. Passive Study Be actively involved in managing the learning process, the mathematics and your study
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 information5 th Grade Common Core State Standards. Flip Book
5 th Grade Common Core State Standards Flip Book This document is intended to show the connections to the Standards of Mathematical Practices for the content standards and to get detailed information at
More informationcopyright 2011 Creative Smarts Inc
copyright 2011 Creative Smarts Inc x Greg Tang s x Greg Tang s x Greg Tang s Great Times TM Series Overview Multiplication Worksheets &'$()!"#$% Multiplication Worksheets Greg Tang s LEARNING the Basic
More informationSupport for Student Literacy
Support for Student Literacy Introduction In today s schools, many students struggle with English language literacy. Some students grow up speaking, reading and/or writing other languages before being
More informationSTUDENTS REASONING IN QUADRATIC EQUATIONS WITH ONE UNKNOWN
STUDENTS REASONING IN QUADRATIC EQUATIONS WITH ONE UNKNOWN M. Gözde Didiş, Sinem Baş, A. Kürşat Erbaş Middle East Technical University This study examined 10 th grade students procedures for solving quadratic
More informationOverview. Essential Questions. Precalculus, Quarter 3, Unit 3.4 Arithmetic Operations With Matrices
Arithmetic Operations With Matrices Overview Number of instruction days: 6 8 (1 day = 53 minutes) Content to Be Learned Use matrices to represent and manipulate data. Perform arithmetic operations with
More informationTeaching And Learning Of Mathematics
Teaching And Learning Of Mathematics D r An a S a v i c, m r S v e t l a n a S t r b a c  S a v i c B e l g r a d e, 11 t h O c t o b e r 2 0 1 2. V I S E R, R I C U M Why we teach mathematics? We organize
More informationHelping Students Develop Confidence to Learn Mathematics
Helping Students Develop Confidence to Learn Mathematics Presentation at the CMC, Southern Section Palm Springs, CA November 6, 2010 Dr. Randy Philipp San Diego State University Alison Williams Fay Elementary
More informationUnit: Psychological Research. Collection of Instructions and Worksheets
Unit: Psychological Research Collection of Instructions and Worksheets INTRODUCTION TO PSYCHOLOGY Name RESEARCH EXPERIMENT (400 points) The goal of this project is to provide you with the experience of
More informationElementary Science Notebooks
Elementary Science Notebooks As you skim through this document, think about purpose: are you deciding whether or not to begin using notebooks in your science instruction? Deciding which way to implement?
More informationTriad Essay. *Learned about myself as a Teacher*
Triad Essay *Learned about myself as a Teacher* Through doing the triads, I learned a lot more than I thought about my teaching. I was surprised to see how hard it is not to jump in and tell someone the
More informationPRACTICE BOOK MATHEMATICS TEST (RESCALED) Graduate Record Examinations. This practice book contains. Become familiar with
This book is provided FREE with test registration by the Graduate Record Examinations Board. Graduate Record Examinations This practice book contains one actual fulllength GRE Mathematics Test (Rescaled)
More informationOverview. Essential Questions. Grade 2 Mathematics, Quarter 4, Unit 4.4 Representing and Interpreting Data Using Picture and Bar Graphs
Grade 2 Mathematics, Quarter 4, Unit 4.4 Representing and Interpreting Data Using Picture and Bar Graphs Overview Number of instruction days: 7 9 (1 day = 90 minutes) Content to Be Learned Draw a picture
More informationMathematics Content Courses for Elementary Teachers
Mathematics Content Courses for Elementary Teachers Sybilla Beckmann Department of Mathematics University of Georgia Massachusetts, March 2008 Sybilla Beckmann (University of Georgia) Mathematics for Elementary
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 informationSupporting your KS2 child s learning in multiplication and division
Our schoolhome agreed mathematics strategies Supporting your KS2 child s learning in multiplication and division Information for Parents and Carers to dip in and out of when necessary Key vocabulary for
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 informationExaminations Page 1 MODULE 2 STUDENT GUIDE TO NOTE TAKING
Examinations Page 1 MODULE 2 STUDENT GUIDE TO NOTE TAKING Examinations Page 2 Why take notes? I have a good memory: It s just short" o s t o f u s c a n r e l a t e t o t h i s q u o t e. H a v e y o u
More informationMathematics. Curriculum Content for Elementary School Mathematics. Fulton County Schools Curriculum Guide for Elementary Schools
Mathematics Philosophy Mathematics permeates all sectors of life and occupies a wellestablished position in curriculum and instruction. Schools must assume responsibility for empowering students with
More information5th Grade. Division.
1 5th Grade Division 2015 11 25 www.njctl.org 2 Division Unit Topics Click on the topic to go to that section Divisibility Rules Patterns in Multiplication and Division Division of Whole Numbers Division
More informationGrades Three though Five Number Talks Based on Number Talks by Sherry Parrish, Math Solutions 2010
Grades Three though Five Number Talks Based on Number Talks by Sherry Parrish, Math Solutions 2010 Number Talks is a tenminute classroom routine included in this year s Scope and Sequence. Kindergarten
More informationOverview. Essential Questions. Grade 5 Mathematics, Quarter 3, Unit 3.2 Multiplying and Dividing With Decimals
Multiplying and Dividing With Decimals Overview Number of instruction days: 9 11 (1 day = 90 minutes) Content to Be Learned Multiply decimals to hundredths. Divide decimals to hundredths. Use models, drawings,
More informationPRACTICE BOOK COMPUTER SCIENCE TEST. Graduate Record Examinations. This practice book contains. Become familiar with. Visit GRE Online at www.gre.
This book is provided FREE with test registration by the Graduate Record Examinations Board. Graduate Record Examinations This practice book contains one actual fulllength GRE Computer Science Test testtaking
More informationBonneygrove Primary School Calculations Policy
Bonneygrove Primary School Calculations Policy Rationale At Bonneygrove, we strongly encourage children to independently use a variety of practical resources to support their learning for each stage of
More information