Multiplication and Divisions

Total Page：16

File Type：pdf, Size：1020Kb

Third Grade Math nd 2 Grading Period Power Objectives: Academic Vocabulary: multiplication array Represent and solve problems involving multiplication and divisor commutative division. (P.O. #1) property Understand properties of multiplication and the distributive property relationship between multiplication and divisions. estimation division factor column (P.O. #2) Multiply and divide with 100. (P.O. #3) repeated addition multiple Solve problems involving the four operations, and identify associative property quotient and explain the patterns in arithmetic. (P.O. #4) rounding row product equation Multiplication and Division Enduring Understandings: Essential Questions: Mathematical operations are used in solving problems In what ways can operations affect numbers? in which a new value is produced from one or more How can different strategies be helpful when solving a values. problem? Algebraic thinking involves choosing, combining, and How does knowing and using algorithms help us to be applying effective strategies for answering questions. Numbers enable us to use the four operations to efficient problem solvers? combine and separate quantities. How are multiplication and addition alike? See below for additional enduring understandings. How are subtraction and division related? See below for additional essential questions. Enduring Understandings: Multiplication is repeated addition, related to division, and can be used to solve story problems. For a given set of numbers, there are relationships that are always true called properties, and these are the rules that govern arithmetic. Multiplication and division have inverse relationships. Essential Questions: Is there more than one way of multiplying to get the same product? How can the same array represent both multiplication and division? How can you use multiplication to help you divide? What are strategies for learning multiplication facts? How does understanding the commutative property help us multiply? How does understanding the distributive property help us multiply large numbers? How can the use of estimation help us make logical problem solving decisions? How does understanding patterns, sequences, and functions help us to solve problems? .

Copy Link

Recommended publications

Lesson 2: the Multiplication of Polynomials
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 2 M1 ALGEBRA II Lesson 2: The Multiplication of Polynomials Student Outcomes . Students develop the distributive property for application to polynomial multiplication. Students connect multiplication of polynomials with multiplication of multi-digit integers. Lesson Notes This lesson begins to address standards A-SSE.A.2 and A-APR.C.4 directly and provides opportunities for students to practice MP.7 and MP.8. The work is scaffolded to allow students to discern patterns in repeated calculations, leading to some general polynomial identities that are explored further in the remaining lessons of this module. As in the last lesson, if students struggle with this lesson, they may need to review concepts covered in previous grades, such as: The connection between area properties and the distributive property: Grade 7, Module 6, Lesson 21. Introduction to the table method of multiplying polynomials: Algebra I, Module 1, Lesson 9. Multiplying polynomials (in the context of quadratics): Algebra I, Module 4, Lessons 1 and 2. Since division is the inverse operation of multiplication, it is important to make sure that your students understand how to multiply polynomials before moving on to division of polynomials in Lesson 3 of this module. In Lesson 3, division is explored using the reverse tabular method, so it is important for students to work through the table diagrams in this lesson to prepare them for the upcoming work. There continues to be a sharp distinction in this curriculum between justification and proof, such as justifying the identity (푎 + 푏)2 = 푎2 + 2푎푏 + 푏 using area properties and proving the identity using the distributive property.
[Show full text]
The Role of the Interval Domain in Modern Exact Real Airthmetic
The Role of the Interval Domain in Modern Exact Real Airthmetic Andrej Bauer Iztok Kavkler Faculty of Mathematics and Physics University of Ljubljana, Slovenia Domains VIII & Computability over Continuous Data Types Novosibirsk, September 2007 Teaching theoreticians a lesson Recently I have been told by an anonymous referee that “Theoreticians do not like to be taught lessons.” and by a friend that “You should stop competing with programmers.” In deﬁance of this advice, I shall talk about the lessons I learned, as a theoretician, in programming exact real arithmetic. The spectrum of real number computation slow fast Formally verified, Cauchy sequences iRRAM extracted from streams of signed digits RealLib proofs floating point Moebius transformtions continued fractions Mathematica "theoretical" "practical" I Common features: I Reals are represented by successive approximations. I Approximations may be computed to any desired accuracy. I State of the art, as far as speed is concerned: I iRRAM by Norbert Muller,¨ I RealLib by Branimir Lambov. What makes iRRAM and ReaLib fast? I Reals are represented by sequences of dyadic intervals (endpoints are rationals of the form m/2k). I The approximating sequences need not be nested chains of intervals. I No guarantee on speed of converge, but arbitrarily fast convergence is possible. I Previous approximations are not stored and not reused when the next approximation is computed. I Each next approximation roughly doubles the amount of work done. The theory behind iRRAM and RealLib I Theoretical models used to design iRRAM and RealLib: I Type Two Effectivity I a version of Real RAM machines I Type I representations I The authors explicitly reject domain theory as a suitable computational model.
[Show full text]
Division by Fractions 6.1.1 - 6.1.4
DIVISION BY FRACTIONS 6.1.1 - 6.1.4 Division by fractions introduces three methods to help students understand how dividing by fractions works. In general, think of division for a problem like 8..,.. 2 as, "In 8, how many groups of 2 are there?" Similarly, ½ + ¼ means, "In ½ , how many fourths are there?" For more information, see the Math Notes boxes in Lessons 7.2 .2 and 7 .2 .4 of the Core Connections, Course 1 text. For additional examples and practice, see the Core Connections, Course 1 Checkpoint 8B materials. The first two examples show how to divide fractions using a diagram. Example 1 Use the rectangular model to divide: ½ + ¼ . Step 1: Using the rectangle, we first divide it into 2 equal pieces. Each piece represents ½. Shade ½ of it. - Step 2: Then divide the original rectangle into four equal pieces. Each section represents ¼ . In the shaded section, ½ , there are 2 fourths. 2 Step 3: Write the equation. Example 2 In ¾ , how many ½ s are there? In ¾ there is one full ½ 2 2 I shaded and half of another Thatis,¾+½=? one (that is half of one half). ]_ ..,_ .l 1 .l So. 4 . 2 = 2 Start with ¾ . 3 4 (one and one-half halves) Parent Guide with Extra Practice © 2011, 2013 CPM Educational Program. All rights reserved. 49 Problems Use the rectangular model to divide. .l ...:... J_ 1 ...:... .l 1. ..,_ l 1 . 1 3 . 6 2. 3. 4. 1 4 . 2 5. 2 3 . 9 Answers l. 8 2. 2 3. 4 one thirds rm I I halves - ~I sixths fourths fourths ~I 11 ~'.¿;¡~:;¿~ ffk] 8 sixths 2 three fourths 4.
[Show full text]
Chapter 2. Multiplication and Division of Whole Numbers in the Last Chapter You Saw That Addition and Subtraction Were Inverse Mathematical Operations
Chapter 2. Multiplication and Division of Whole Numbers In the last chapter you saw that addition and subtraction were inverse mathematical operations. For example, a pay raise of 50 cents an hour is the opposite of a 50 cents an hour pay cut. When you have completed this chapter, you’ll understand that multiplication and division are also inverse math- ematical operations. 2.1 Multiplication with Whole Numbers The Multiplication Table Learning the multiplication table shown below is a basic skill that must be mastered. Do you have to memorize this table? Yes! Can’t you just use a calculator? No! You must know this table by heart to be able to multiply numbers, to do division, and to do algebra. To be blunt, until you memorize this entire table, you won’t be able to progress further than this page. MULTIPLICATION TABLE ϫ 012 345 67 89101112 0 000 000LEARNING 00 000 00 1 012 345 67 89101112 2 024 681012Copy14 16 18 20 22 24 3 036 9121518212427303336 4 0481216 20 24 28 32 36 40 44 48 5051015202530354045505560 6061218243036424854606672Distribute 7071421283542495663707784 8081624324048566472808896 90918273HAWKESReview645546372819099108 10 0 10 20 30 40 50 60 70 80 90 100 110 120 ©11 0 11 22 33 44NOT 55 66 77 88 99 110 121 132 12 0 12 24 36 48 60 72 84 96 108 120 132 144 Do Let’s get a couple of things out of the way. First, any number times 0 is 0. When we multiply two numbers, we call our answer the product of those two numbers.
[Show full text]
Grade 7/8 Math Circles the Scale of Numbers Introduction
Faculty of Mathematics Centre for Education in Waterloo, Ontario N2L 3G1 Mathematics and Computing Grade 7/8 Math Circles November 21/22/23, 2017 The Scale of Numbers Introduction Last week we quickly took a look at scientific notation, which is one way we can write down really big numbers. We can also use scientific notation to write very small numbers. 1 × 103 = 1; 000 1 × 102 = 100 1 × 101 = 10 1 × 100 = 1 1 × 10−1 = 0:1 1 × 10−2 = 0:01 1 × 10−3 = 0:001 As you can see above, every time the value of the exponent decreases, the number gets smaller by a factor of 10. This pattern continues even into negative exponent values! Another way of picturing negative exponents is as a division by a positive exponent. 1 10−6 = = 0:000001 106 In this lesson we will be looking at some famous, interesting, or important small numbers, and begin slowly working our way up to the biggest numbers ever used in mathematics! Obviously we can come up with any arbitrary number that is either extremely small or extremely large, but the purpose of this lesson is to only look at numbers with some kind of mathematical or scientific significance. 1 Extremely Small Numbers 1. Zero • Zero or `0' is the number that represents nothingness. It is the number with the smallest magnitude. • Zero only began being used as a number around the year 500. Before this, ancient mathematicians struggled with the concept of `nothing' being `something'. 2. Planck's Constant This is the smallest number that we will be looking at today other than zero.
[Show full text]
The Notion Of" Unimaginable Numbers" in Computational Number Theory
Beyond Knuth’s notation for “Unimaginable Numbers” within computational number theory Antonino Leonardis1 - Gianfranco d’Atri2 - Fabio Caldarola3 1 Department of Mathematics and Computer Science, University of Calabria Arcavacata di Rende, Italy e-mail: [email protected] 2 Department of Mathematics and Computer Science, University of Calabria Arcavacata di Rende, Italy 3 Department of Mathematics and Computer Science, University of Calabria Arcavacata di Rende, Italy e-mail: [email protected] Abstract Literature considers under the name unimaginable numbers any positive in- teger going beyond any physical application, with this being more of a vague description of what we are talking about rather than an actual mathemati- cal definition (it is indeed used in many sources without a proper definition). This simply means that research in this topic must always consider shortened representations, usually involving recursion, to even being able to describe such numbers. One of the most known methodologies to conceive such numbers is using hyper-operations, that is a sequence of binary functions defined recursively starting from the usual chain: addition - multiplication - exponentiation. arXiv:1901.05372v2 [cs.LO] 12 Mar 2019 The most important notations to represent such hyper-operations have been considered by Knuth, Goodstein, Ackermann and Conway as described in this work’s introduction. Within this work we will give an axiomatic setup for this topic, and then try to find on one hand other ways to represent unimaginable numbers, as well as on the other hand applications to computer science, where the algorith- mic nature of representations and the increased computation capabilities of 1 computers give the perfect field to develop further the topic, exploring some possibilities to effectively operate with such big numbers.
[Show full text]
Fast Integer Division – a Differentiated Offering from C2000 Product Family
Application Report SPRACN6–July 2019 Fast Integer Division – A Differentiated Offering From C2000™ Product Family Prasanth Viswanathan Pillai, Himanshu Chaudhary, Aravindhan Karuppiah, Alex Tessarolo ABSTRACT This application report provides an overview of the different division and modulo (remainder) functions and its associated properties. Later, the document describes how the different division functions can be implemented using the C28x ISA and intrinsics supported by the compiler. Contents 1 Introduction ................................................................................................................... 2 2 Different Division Functions ................................................................................................ 2 3 Intrinsic Support Through TI C2000 Compiler ........................................................................... 4 4 Cycle Count................................................................................................................... 6 5 Summary...................................................................................................................... 6 6 References ................................................................................................................... 6 List of Figures 1 Truncated Division Function................................................................................................ 2 2 Floored Division Function................................................................................................... 3 3 Euclidean
[Show full text]
Multiplication Fact Strategies Assessment Directions and Analysis
Multiplication Fact Strategies Wichita Public Schools Curriculum and Instructional Design Mathematics Revised 2014 KCCRS version Table of Contents Introduction Page Research Connections (Strategies) 3 Making Meaning for Operations 7 Assessment 9 Tools 13 Doubles 23 Fives 31 Zeroes and Ones 35 Strategy Focus Review 41 Tens 45 Nines 48 Squared Numbers 54 Strategy Focus Review 59 Double and Double Again 64 Double and One More Set 69 Half and Then Double 74 Strategy Focus Review 80 Related Equations (fact families) 82 Practice and Review 92 Wichita Public Schools 2014 2 Research Connections Where Do Fact Strategies Fit In? Adapted from Randall Charles Fact strategies are considered a crucial second phase in a three-phase program for teaching students basic math facts. The first phase is concept learning. Here, the goal is for students to understand the meanings of multiplication and division. In this phase, students focus on actions (i.e. “groups of”, “equal parts”, “building arrays”) that relate to multiplication and division concepts. An important instructional bridge that is often neglected between concept learning and memorization is the second phase, fact strategies. There are two goals in this phase. First, students need to recognize there are clusters of multiplication and division facts that relate in certain ways. Second, students need to understand those relationships. These lessons are designed to assist with the second phase of this process. If you have students that are not ready, you will need to address the first phase of concept learning. The third phase is memorization of the basic facts. Here the goal is for students to master products and quotients so they can recall them efficiently and accurately, and retain them over time.
[Show full text]
Matrix Multiplication. Diagonal Matrices. Inverse Matrix. Matrices
MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. Matrices Definition. An m-by-n matrix is a rectangular array of numbers that has m rows and n columns: a11 a12 ... a1n a21 a22 ... a2n . .. . am1 am2 ... amn Notation: A = (aij )1≤i≤n, 1≤j≤m or simply A = (aij ) if the dimensions are known. Matrix algebra: linear operations Addition: two matrices of the same dimensions can be added by adding their corresponding entries. Scalar multiplication: to multiply a matrix A by a scalar r, one multiplies each entry of A by r. Zero matrix O: all entries are zeros. Negative: −A is defined as (−1)A. Subtraction: A − B is defined as A + (−B). As far as the linear operations are concerned, the m×n matrices can be regarded as mn-dimensional vectors. Properties of linear operations (A + B) + C = A + (B + C) A + B = B + A A + O = O + A = A A + (−A) = (−A) + A = O r(sA) = (rs)A r(A + B) = rA + rB (r + s)A = rA + sA 1A = A 0A = O Dot product Definition. The dot product of n-dimensional vectors x = (x1, x2,..., xn) and y = (y1, y2,..., yn) is a scalar n x · y = x1y1 + x2y2 + ··· + xnyn = xk yk . Xk=1 The dot product is also called the scalar product. Matrix multiplication The product of matrices A and B is defined if the number of columns in A matches the number of rows in B. Definition. Let A = (aik ) be an m×n matrix and B = (bkj ) be an n×p matrix.
[Show full text]
AUTHOR Multiplication and Division. Mathematics-Methods
DOCUMENT RESUME ED 221 338 SE 035 008, AUTHOR ,LeBlanc, John F.; And Others TITLE Multiplication and Division. Mathematics-Methods 12rogram Unipt. INSTITUTION Indiana ,Univ., Bloomington. Mathematics Education !Development Center. , SPONS AGENCY ,National Science Foundation, Washington, D.C. REPORT NO :ISBN-0-201-14610-x PUB DATE 76 . GRANT. NSF-GY-9293 i 153p.; For related documents, see SE 035 006-018. NOTE . ,. EDRS PRICE 'MF01 Plus Postage. PC Not Available from EDRS. DESCRIPTORS iAlgorithms; College Mathematics; *Division; Elementary Education; *Elementary School Mathematics; 'Elementary School Teachers; Higher Education; 'Instructional1 Materials; Learning Activities; :*Mathematics Instruction; *Multiplication; *Preservice Teacher Education; Problem Solving; :Supplementary Reading Materials; Textbooks; ;Undergraduate Study; Workbooks IDENTIFIERS i*Mathematics Methods Program ABSTRACT This unit is 1 of 12 developed for the university classroom portign_of the Mathematics-Methods Program(MMP), created by the Indiana pniversity Mathematics EducationDevelopment Center (MEDC) as an innovative program for the mathematics training of prospective elenientary school teachers (PSTs). Each unit iswritten in an activity format that involves the PST in doing mathematicswith an eye towardaiplication of that mathematics in the elementary school. This do ument is one of four units that are devoted tothe basic number wo k in the elementary school. In addition to an introduction to the unit, the text has sections on the conceptual development of nultiplication
[Show full text]
Basic Math Quick Reference Ebook
This file is distributed FREE OF CHARGE by the publisher Quick Reference Handbooks and the author. Quick Reference eBOOK Click on The math facts listed in this eBook are explained with Contents or Examples, Notes, Tips, & Illustrations Index in the in the left Basic Math Quick Reference Handbook. panel ISBN: 978-0-615-27390-7 to locate a topic. Peter J. Mitas Quick Reference Handbooks Facts from the Basic Math Quick Reference Handbook Contents Click a CHAPTER TITLE to jump to a page in the Contents: Whole Numbers Probability and Statistics Fractions Geometry and Measurement Decimal Numbers Positive and Negative Numbers Universal Number Concepts Algebra Ratios, Proportions, and Percents … then click a LINE IN THE CONTENTS to jump to a topic. Whole Numbers 7 Natural Numbers and Whole Numbers ............................ 7 Digits and Numerals ........................................................ 7 Place Value Notation ....................................................... 7 Rounding a Whole Number ............................................. 8 Operations and Operators ............................................... 8 Adding Whole Numbers................................................... 9 Subtracting Whole Numbers .......................................... 10 Multiplying Whole Numbers ........................................... 11 Dividing Whole Numbers ............................................... 12 Divisibility Rules ............................................................ 13 Multiples of a Whole Number .......................................
[Show full text]
Episodes in the History of Modern Algebra (1800–1950)
HISTORY OF MATHEMATICS • VOLUME 32 Episodes in the History of Modern Algebra (1800–1950) Jeremy J. Gray Karen Hunger Parshall Editors American Mathematical Society • London Mathematical Society Episodes in the History of Modern Algebra (1800–1950) https://doi.org/10.1090/hmath/032 HISTORY OF MATHEMATICS v VOLUME 32 Episodes in the History of Modern Algebra (1800–1950) Jeremy J. Gray Karen Hunger Parshall Editors Editorial Board American Mathematical Society London Mathematical Society Joseph W. Dauben Alex D. D. Craik Peter Duren Jeremy J. Gray Karen Parshall, Chair Robin Wilson, Chair MichaelI.Rosen 2000 Mathematics Subject Classiﬁcation. Primary 01A55, 01A60, 01A70, 01A72, 01A73, 01A74, 01A80. For additional information and updates on this book, visit www.ams.org/bookpages/hmath-32 Library of Congress Cataloging-in-Publication Data Episodes in the history of modern algebra (1800–1950) / Jeremy J. Gray and Karen Hunger Parshall, editors. p. cm. Includes bibliographical references and index. ISBN-13: 978-0-8218-4343-7 (alk. paper) ISBN-10: 0-8218-4343-5 (alk. paper) 1. Algebra—History. I. Gray, Jeremy, 1947– II. Parshall, Karen Hunger, 1955– QA151.E65 2007 512.009—dc22 2007060683 AMS softcover ISBN: 978-0-8218-6904-8 Copying and reprinting. Individual readers of this publication, and nonproﬁt libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society.
[Show full text]