Division Short and Long

Total Page：16

File Type：pdf, Size：1020Kb

Division Short and Long This presentation is intended to be used by the parents and students of Mrs. Nicholas SRA Connecting Math Concepts Math program. It is not intended to be a substitute for a research based program. Division Terms 9 Quotient Divisor 7 63 Dividend Short Division basic facts • The answer is part Examples: of the basic fact 7 6 family 8 56 4 24 • No remainder 5 3 2 10 6 18 Short Division with Remainder • Dividend is not a basic fact. • Steps to solve: 1. Determine how close the divisor can get to the dividend without going over. Multiplication charts can help 6 20 2. Write the answer below the dividend. 6 20 18 3. The quotient is the second part of the basic fact family. 3 6 20 *Basic fact: 6 x 3 =18 18 4. Subtract the basic fact answer from the original quotient. 3 6 20 18 5. The subtract answer is the remainder. 3 R 2 6 20 18 Long Division • Long division is a division problem that has multiple digits in the quotient. • Steps to solve are same as short division. You repeat the 4 steps until all digits in the dividend are solved. Long Division Example 2 6 8 8 1. Ask: Can the divisor into the first digit. 2 6 8 8 • Yes 3 4 4 2. Follow steps 1-4 under short division. 2 6 8 8 3. Continue until all digits 6 4 4 have an answer over the dividend. Long Division Special circumstances…that will occur a lot! 1. Ask: Can the divisor into the first 1 digit. 5 5 4 5 • Yes, continue with steps 1-4 5 2. Next digit, start again. Can the divisor go in the next digit. • No. Put a zero above that 1 0 9 number and underline the next digit and ask the 5 5 4 5 question again. 5 45 • Now the answer is yes. Continue with steps 1-4. Zero • Rule to live by: 3 0 4 Anything divided 2 6 0 8 by zero is zero More Long Division • All division rule reviewed so far still apply; however sometimes you will have subtract many different times. • The answer to the subtraction problem is moved to the next digit in the dividend to create a number. • The answer to the final problem is the remainder. Example 1 1 0 7 Step 1 4 4 3 5 9 Step 2 4 4 3 8 9 35 1 0 7 Step 4 1 0 7 9 4 4 3 8 39 Step 5 3 5 4 4 3 8 39 Bring the answer to subtraction 35 36 problem to the next digit in the division problem 1 0 7 9 r 3 Step 6 4 4 3 8 39 35 36 Final reminder • All number in the quotient need to line up with the numbers in the dividend. • Look back at the example problems throughout the presentation..

Copy Link

Recommended publications

Lesson 19: the Euclidean Algorithm As an Application of the Long Division Algorithm
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 19 6•2 Lesson 19: The Euclidean Algorithm as an Application of the Long Division Algorithm Student Outcomes . Students explore and discover that Euclid’s Algorithm is a more efficient means to finding the greatest common factor of larger numbers and determine that Euclid’s Algorithm is based on long division. Lesson Notes MP.7 Students look for and make use of structure, connecting long division to Euclid’s Algorithm. Students look for and express regularity in repeated calculations leading to finding the greatest common factor of a pair of numbers. These steps are contained in the Student Materials and should be reproduced, so they can be displayed throughout the lesson: Euclid’s Algorithm is used to find the greatest common factor (GCF) of two whole numbers. MP.8 1. Divide the larger of the two numbers by the smaller one. 2. If there is a remainder, divide it into the divisor. 3. Continue dividing the last divisor by the last remainder until the remainder is zero. 4. The final divisor is the GCF of the original pair of numbers. In application, the algorithm can be used to find the side length of the largest square that can be used to completely fill a rectangle so that there is no overlap or gaps. Classwork Opening (5 minutes) Lesson 18 Problem Set can be discussed before going on to this lesson. Lesson 19: The Euclidean Algorithm as an Application of the Long Division Algorithm 178 Date: 4/1/14 This work is licensed under a © 2013 Common Core, Inc.
[Show full text]
Three Methods of Finding Remainder on the Operation of Modular
Zin Mar Win, Khin Mar Cho; International Journal of Advance Research and Development (Volume 5, Issue 5) Available online at: www.ijarnd.com Three methods of finding remainder on the operation of modular exponentiation by using calculator 1Zin Mar Win, 2Khin Mar Cho 1University of Computer Studies, Myitkyina, Myanmar 2University of Computer Studies, Pyay, Myanmar ABSTRACT The primary purpose of this paper is that the contents of the paper were to become a learning aid for the learners. Learning aids enhance one's learning abilities and help to increase one's learning potential. They may include books, diagram, computer, recordings, notes, strategies, or any other appropriate items. In this paper we would review the types of modulo operations and represent the knowledge which we have been known with the easiest ways. The modulo operation is finding of the remainder when dividing. The modulus operator abbreviated “mod” or “%” in many programming languages is useful in a variety of circumstances. It is commonly used to take a randomly generated number and reduce that number to a random number on a smaller range, and it can also quickly tell us if one number is a factor of another. If we wanted to know if a number was odd or even, we could use modulus to quickly tell us by asking for the remainder of the number when divided by 2. Modular exponentiation is a type of exponentiation performed over a modulus. The operation of modular exponentiation calculates the remainder c when an integer b rose to the eth power, bᵉ, is divided by a positive integer m such as c = be mod m [1].
[Show full text]
1.3 Division of Polynomials; Remainder and Factor Theorems
1-32 CHAPTER 1. POLYNOMIAL AND RATIONAL FUNCTIONS 1.3 Division of Polynomials; Remainder and Factor Theorems Objectives • Perform long division of polynomials • Perform synthetic division of polynomials • Apply remainder and factor theorems In previous courses, you may have learned how to factor polynomials using various techniques. Many of these techniques apply only to special kinds of polynomial expressions. For example, the the previous two sections of this chapter dealt only with polynomials that could easily be factored to find the zeros and x-intercepts. In order to be able to find zeros and x-intercepts of polynomials which cannot readily be factored, we first introduce long division of polynomials. We will then use the long division algorithm to make a general statement about factors of a polynomial. Long division of polynomials Long division of polynomials is similar to long division of numbers. When divid- ing polynomials, we obtain a quotient and a remainder. Just as with numbers, if a remainder is 0, then the divisor is a factor of the dividend. Example 1 Determining Factors by Division Divide to determine whether x − 1 is a factor of x2 − 3x +2. Solution Here x2 − 3x +2is the dividend and x − 1 is the divisor. Step 1 Set up the division as follows: divisor → x − 1 x2−3x+2 ← dividend Step 2 Divide the leading term of the dividend (x2) by the leading term of the divisor (x). The result (x)is the first term of the quotient, as illustrated below. x ← first term of quotient x − 1 x2 −3x+2 1.3.
[Show full text]
Version 0.5.1 of 5 March 2010
Modern Computer Arithmetic Richard P. Brent and Paul Zimmermann Version 0.5.1 of 5 March 2010 iii Copyright c 2003-2010 Richard P. Brent and Paul Zimmermann ° This electronic version is distributed under the terms and conditions of the Creative Commons license “Attribution-Noncommercial-No Derivative Works 3.0”. You are free to copy, distribute and transmit this book under the following conditions: Attribution. You must attribute the work in the manner speciﬁed by the • author or licensor (but not in any way that suggests that they endorse you or your use of the work). Noncommercial. You may not use this work for commercial purposes. • No Derivative Works. You may not alter, transform, or build upon this • work. For any reuse or distribution, you must make clear to others the license terms of this work. The best way to do this is with a link to the web page below. Any of the above conditions can be waived if you get permission from the copyright holder. Nothing in this license impairs or restricts the author’s moral rights. For more information about the license, visit http://creativecommons.org/licenses/by-nc-nd/3.0/ Contents Preface page ix Acknowledgements xi Notation xiii 1 Integer Arithmetic 1 1.1 Representation and Notations 1 1.2 Addition and Subtraction 2 1.3 Multiplication 3 1.3.1 Naive Multiplication 4 1.3.2 Karatsuba’s Algorithm 5 1.3.3 Toom-Cook Multiplication 6 1.3.4 Use of the Fast Fourier Transform (FFT) 8 1.3.5 Unbalanced Multiplication 8 1.3.6 Squaring 11 1.3.7 Multiplication by a Constant 13 1.4 Division 14 1.4.1 Naive
[Show full text]
Primality Testing for Beginners
STUDENT MATHEMATICAL LIBRARY Volume 70 Primality Testing for Beginners Lasse Rempe-Gillen Rebecca Waldecker http://dx.doi.org/10.1090/stml/070 Primality Testing for Beginners STUDENT MATHEMATICAL LIBRARY Volume 70 Primality Testing for Beginners Lasse Rempe-Gillen Rebecca Waldecker American Mathematical Society Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell Gerald B. Folland (Chair) Serge Tabachnikov The cover illustration is a variant of the Sieve of Eratosthenes (Sec- tion 1.5), showing the integers from 1 to 2704 colored by the number of their prime factors, including repeats. The illustration was created us- ing MATLAB. The back cover shows a phase plot of the Riemann zeta function (see Appendix A), which appears courtesy of Elias Wegert (www.visual.wegert.com). 2010 Mathematics Subject Classification. Primary 11-01, 11-02, 11Axx, 11Y11, 11Y16. For additional information and updates on this book, visit www.ams.org/bookpages/stml-70 Library of Congress Cataloging-in-Publication Data Rempe-Gillen, Lasse, 1978– author. [Primzahltests für Einsteiger. English] Primality testing for beginners / Lasse Rempe-Gillen, Rebecca Waldecker. pages cm. — (Student mathematical library ; volume 70) Translation of: Primzahltests für Einsteiger : Zahlentheorie - Algorithmik - Kryptographie. Includes bibliographical references and index. ISBN 978-0-8218-9883-3 (alk. paper) 1. Number theory. I. Waldecker, Rebecca, 1979– author. II. Title. QA241.R45813 2014 512.72—dc23 2013032423 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.
[Show full text]
Dividing Whole Numbers
Dividing Whole Numbers INTRODUCTION Consider the following situations: 1. Edgar brought home 72 donut holes for the 9 children at his daughter’s slumber party. How many will each child receive if the donut holes are divided equally among all nine? 2. Three friends, Gloria, Jan, and Mie-Yun won a small lottery prize of $1,450. If they split it equally among all three, how much will each get? 3. Shawntee just purchased a used car. She has agreed to pay a total of $6,200 over a 36- month period. What will her monthly payments be? 4. 195 people are planning to attend an awards banquet. Ruben is in charge of the table rental. If each table seats 8, how many tables are needed so that everyone has a seat? Each of these problems can be answered using division. As you’ll recall from Section 1.2, the result of division is called the quotient. Here are the parts of a division problem: Standard form: dividend ÷ divisor = quotient Long division form: dividend Fraction form: divisor = quotient We use these words—dividend, divisor, and quotient—throughout this section, so it’s important that you become familiar with them. You’ll often see this little diagram as a continual reminder of their proper placement. Here is how a division problem is read: In the standard form: 15 ÷ 5 is read “15 divided by 5.” In the long division form: 5 15 is read “5 divided into 15.” 15 In the fraction form: 5 is read “15 divided by 5.” Dividing Whole Numbers page 1 Example 1: In this division problem, identify the dividend, divisor, and quotient.
[Show full text]
Grade 7/8 Math Circles Modular Arithmetic the Modulus Operator
Faculty of Mathematics Centre for Education in Waterloo, Ontario N2L 3G1 Mathematics and Computing Grade 7/8 Math Circles April 3, 2014 Modular Arithmetic The Modulus Operator The modulo operator has symbol \mod", is written as A mod N, and is read \A modulo N" or "A mod N". • A is our - what is being divided (just like in division) • N is our (the plural is ) When we divide two numbers A and N, we get a quotient and a remainder. • When we ask for A ÷ N, we look for the quotient of the division. • When we ask for A mod N, we are looking for the remainder of the division. • The remainder A mod N is always an integer between 0 and N − 1 (inclusive) Examples 1. 0 mod 4 = 0 since 0 ÷ 4 = 0 remainder 0 2. 1 mod 4 = 1 since 1 ÷ 4 = 0 remainder 1 3. 2 mod 4 = 2 since 2 ÷ 4 = 0 remainder 2 4. 3 mod 4 = 3 since 3 ÷ 4 = 0 remainder 3 5. 4 mod 4 = since 4 ÷ 4 = 1 remainder 6. 5 mod 4 = since 5 ÷ 4 = 1 remainder 7. 6 mod 4 = since 6 ÷ 4 = 1 remainder 8. 15 mod 4 = since 15 ÷ 4 = 3 remainder 9.* −15 mod 4 = since −15 ÷ 4 = −4 remainder Using a Calculator For really large numbers and/or moduli, sometimes we can use calculators to quickly ﬁnd A mod N. This method only works if A ≥ 0. Example: Find 373 mod 6 1. Divide 373 by 6 ! 2. Round the number you got above down to the nearest integer ! 3.
[Show full text]
So You Think You Can Divide?
So You Think You Can Divide? A History of Division Stephen Lucas Department of Mathematics and Statistics James Madison University, Harrisonburg VA October 10, 2011 Tobias Dantzig: Number (1930, p26) “There is a story of a German merchant of the fifteenth century, which I have not succeeded in authenticating, but it is so characteristic of the situation then existing that I cannot resist the temptation of telling it. It appears that the merchant had a son whom he desired to give an advanced commercial education. He appealed to a prominent professor of a university for advice as to where he should send his son. The reply was that if the mathematical curriculum of the young man was to be confined to adding and subtracting, he perhaps could obtain the instruction in a German university; but the art of multiplying and dividing, he continued, had been greatly developed in Italy, which in his opinion was the only country where such advanced instruction could be obtained.” Ancient Techniques Positional Notation Division Yielding Decimals Outline Ancient Techniques Division Yielding Decimals Definitions Integer Division Successive Subtraction Modern Division Doubling Multiply by Reciprocal Geometry Iteration – Newton Positional Notation Iteration – Goldschmidt Iteration – EDSAC Positional Definition Galley or Scratch Factor Napier’s Rods and the “Modern” method Short Division and Genaille’s Rods Double Division Stephen Lucas So You Think You Can Divide? Ancient Techniques Positional Notation Division Yielding Decimals Definitions If a and b are natural numbers and a = qb + r, where q is a nonnegative integer and r is an integer satisfying 0 ≤ r < b, then q is the quotient and r is the remainder after integer division.
[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]
Decimal Long Division EM3TLG1 G5 466Z-NEW.Qx 6/20/08 11:42 AM Page 557
EM3TLG1_G5_466Z-NEW.qx 6/20/08 11:42 AM Page 556 JE PRO CT Objective To extend the long division algorithm to problems in which both the divisor and the dividend are decimals. 1 Doing the Project materials Recommended Use During or after Lesson 4-6 and Project 5. ٗ Math Journal, p. 16 ,Key Activities ٗ Student Reference Book Students explore the meaning of division by a decimal and extend long division to pp. 37, 54G, 54H, and 60 decimal divisors. Key Concepts and Skills • Use long division to solve division problems with decimal divisors. [Operations and Computation Goal 3] • Multiply numbers by powers of 10. [Operations and Computation Goal 3] • Use the Multiplication Rule to find equivalent fractions. [Number and Numeration Goal 5] • Explore the meaning of division by a decimal. [Operations and Computation Goal 7] Key Vocabulary decimal divisors • dividend • divisor 2 Extending the Project materials Students express the remainder in a division problem as a whole number, a fraction, an ٗ Math Journal, p. 17 exact decimal, and a decimal rounded to the nearest hundredth. ٗ Student Reference Book, p. 54I Technology See the iTLG. 466Z Project 14 Decimal Long Division EM3TLG1_G5_466Z-NEW.qx 6/20/08 11:42 AM Page 557 Student Page 1 Doing the Project Date PROJECT 14 Dividing with Decimal Divisors WHOLE-CLASS 1. Draw lines to connect each number model with the number story that fits it best. Number Model Number Story ▼ Exploring Meanings for DISCUSSION What is the area of a rectangle 1.75 m by ?cm 50 0.10 ء Decimal Division 2 Sales tax is 10%.
[Show full text]
Basics of Math 1 Logic and Sets the Statement a Is True, B Is False
Basics of math 1 Logic and sets The statement a is true, b is false. Both statements are false. 9000086601 (level 1): Let a and b be two sentences in the sense of mathematical logic. It is known that the composite statement 9000086604 (level 1): Let a and b be two sentences in the sense of mathematical :(a _ b) logic. It is known that the composite statement is true. For each a and b determine whether it is true or false. :(a ^ :b) is false. For each a and b determine whether it is true or false. Both statements are false. The statement a is true, b is false. Both statements are true. The statement a is true, b is false. Both statements are true. The statement a is false, b is true. The statement a is false, b is true. Both statements are false. 9000086602 (level 1): Let a and b be two sentences in the sense of mathematical logic. It is known that the composite statement 9000086605 (level 1): Let a and b be two sentences in the sense of mathematical :a _ b logic. It is known that the composite statement is false. For each a and b determine whether it is true or false. :a =):b The statement a is true, b is false. is false. For each a and b determine whether it is true or false. The statement a is false, b is true. Both statements are true. The statement a is false, b is true. Both statements are true. The statement a is true, b is false.
[Show full text]
Faster Chinese Remaindering Joris Van Der Hoeven
Faster Chinese remaindering Joris van der Hoeven To cite this version: Joris van der Hoeven. Faster Chinese remaindering. 2016. hal-01403810 HAL Id: hal-01403810 https://hal.archives-ouvertes.fr/hal-01403810 Preprint submitted on 27 Nov 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Faster Chinese remaindering Joris van der Hoeven Laboratoire d'informatique, UMR 7161 CNRS Campus de l'École polytechnique 1, rue Honoré d'Estienne d'Orves Bâtiment Alan Turing, CS35003 91120 Palaiseau November 27, 2016 The Chinese remainder theorem is a key tool for the design of ecient multi-modular algorithms. In this paper, we study the case when the moduli m1; :::; m` are xed and can even be chosen by the user. Through an appropriate use of the technique of FFT-trading, we will show that this assumption allows for the gain of an asymptotic factor O(log log `) in the complexity of Chinese remaindering. For small `, we will also show how to choose gentle moduli that allow for further gains at the other end. The multiplication of integer matrices is one typical application where we expect practical gains for various common matrix dimensions and integer bitsizes.
[Show full text]