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# Logic of Arithmetic 22M 006

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This 12 page Class Notes was uploaded by Virgil Wyman on Friday October 23, 2015. The Class Notes belongs to 22M 006 at University of Iowa taught by Walter Seaman in Fall. Since its upload, it has received 24 views. For similar materials see /class/227989/22m-006-university-of-iowa in Mathematics (M) at University of Iowa.

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Date Created: 10/23/15

Tentative Review Sheet for Fall 2008 Test 2 On 230 pm320 pm in SHAM TOPICS 1 Addition and subtraction algorithms including addition and subtraction in a base different from 10 Multiplication and division defmitions and conceptual models set models number line models rectangular array model for multiplication partitive measurement and missing factors models for division multiple algorithmic approaches such as expanded notation intermediate algorithms duplation lattice methods for multiplication and expanded notation and scaffold type methods for division sections 32 42 43 including multiplication computations in other bases but no division in other bases 2 The number line definition of fractions and its uses in showing equivalence and ordering and section 61 there may be problems on addition and subtraction of fractions the number line definitions of these operations and how they are computed 3 Divisibility factors multiples primes composites prime power representations of natural numbers divisibility tests the total number of factors of a given number GCF LCM sections 51 52 story problem type applications of GCF and LCM e g page 223 20 Those applications may be like the examples given in the class notes on GCF eg common measurements or amounts of objects and LCM eg calendar type problems You may bring with you one 1 8 12 by 11 inch sheet of paper with any of your hand written notes written by you formulas problems etc on them which you think will be useful for taking test 2 You may bring with you a calculator but not a lap top computer or any other type of computer The number of problems on the test will be determined by the level of dif culty and the time available It will have six 6 or fewer separate numbered problems some of which may have parts Test questions will be analogous to the types of questions which have occurred in the inclass tests and some homework and discussion section problems we have had There will be some computational problems and some problems in which you are to give short written answers There might also be some multiple choice questions Here is a very brief summary of topics we have covered since test 1 and for which you will be responsible on test 2 Chapter 3 Whole numbers multiplication and division in the whole numbers You should be prepared to explain any of the set models the number line models or additional conceptual models of these operations in the whole numbers which have occurred in the text and or classes You should be able to give a precise mathematical answer to the question Why division by 0 is unde ned Addition and subtraction algorithms including addition and subtraction in a base different from 10 Multiplication and divisiondefinitions and conceptual models set models number line models rectangular array model for multiplication partitive measurement and missing factors models for division Chapter 4 Whole number computations you will be responsible for knowing how to use the different algorithms for multiplication and division which we have spent a fair amount of time covering during the class Multiple algorithmic approaches such as expanded notation intermediate algorithms duplation lattice methods for multiplication and expanded notation and scaffoldtype methods for division sections 32 42 43 including addition subtraction and multiplication computations in other bases but no division in other bases Recall this table We use the following abbreviations for references below AIU Adding It Up Helping Children Learn Mathematics MBP Mathematics For Elementary Teachers A Contemporary Approach Sixth Edition by Musser Burger and Peterson LDT Mathematical Reasoning for Elementary Teachers Second Edition by Long and DeTemple Addition al orithms 39 quot al orithm s 39 39 39 quot 39 al orithms Division al orithms The common US The common US The common US algorithm The common US algorithm MBP 42 algorithm MBP 42 MBP 42 LDT 34 algorithm LDT 34 MBP 42 Regrouping carrying on Various methods and Intermediate algorithm aka Expanded or repeated the bottom AIU page 202 notations for instructional algorithm subtraction algorithm method B regroupingexchanging Expanded notation LDT 34 method aka borrowing MBP 42 scaffoldtype method LDT 33 AIU p 205 LDT 34 MBP 42 AIU p 211 Intermediate algorithm Equaladditions or Duplation method LDT 34 Scaffoldtype a ka in tructional addends algorithm MBP MBP problem set B exercise methods intermediate set notation 12 4 MBP s or LDT 33 MBP problem the missing addend 34 MBP problem set B LDT 34 set B exercise 4 method MBP p MBP exercise 18 problem set B exercise area 0139 You do not need to memorize the names of these methods although that would probably be useful If you are asked to use a method you will be reminded what that method is You MAY be asked on test 2 to perform the arithmetic operations for addition subtraction or multiplication in bases different than 10 For addition and subtraction you may be asked to use a non standard algorithm in a base different from 10 see review problems 14 and 15 However you will not get questions in which you perform division in a base different from 10 Chapter 6 Fractions section 61 and our number line de nition of fractions You are responsible for knowing the starting number line de nitions of fractions quotients of positive whole numbers with nonzero denominator You will be responsible for knowing how to explain the equivalence of fractions eg 3 10 15 50 using this starting number line de nition of fractions You should be able to describe how to take the number line picturedefinition for for example 3 10 and using that number line and picture show why it s the same dot as the one you get from the number line defn of 15 50 And conversely you should be able to describe how to take the number line picturedefinition for 15 50 and using that number line and picture show why it s the same dot as the one you get from the number line defn of 3 10 You should be able to give such explanations for other equivalent fractions too There may be problems on addition subtraction of fractions the number line de nitions of these operations and how they are computed There may be problems on multiplication and division of fractions These will use our definitions given in the class You should be able to explain how to interpret the division of one fraction by another fraction in terms of the repeated subtraction model of division That explanation was the topic of discussion in lecture and was the subject of the Lecture Writing Assignment the class worked on in the Wed 110508 meeting Chapter 5 Number Theory You are responsible for knowing the definitions of divisibility factor divisor and prime and composite natural numbers You should know how to use the divisibility tests for 2 3 5 9 and 10 You are responsible for knowing how to factor natural numbers into products of primes and count the total number of factors a natural number has You are responsible for knowing how to compute GCF and LCM for two numbers and give descriptions in simple terms of what the GCD and LCM mean You should be able to solve the type of story problems involving GCF andor LCM which we have as text problems and in discussions and lectures PRACTICE PROBLEMS More may be added later Included below are some practice problems The solutions are given below to the problems You should be able to solve these problems without looking at the answers or the book but you can look at your handwritten sheet of notes These problems are representative of the level of difficulty of the questions you might see on test 2 But there are a lot more practice problems than could reasonably be asked in a onehour minute time period There are some topics we have covered which you need to know about for the test 2 which are not covered in these practice problems The practice problems are only a sample of some ofthe topics which may be covered on test 2 1 If a is a whole number give reasons why a 0 is not defined as a whole number at least if one attempts to use the missing factor definition of division Your reasons should be like those in the text or like those discussed in class It is not enough to just so aO is quotundefinedquot or quotnot definedquot Your task is to explain WHY it cannot BE defined as a whole number 3 Give the number line model de nition of a fraction a b with a and b positive integers Explain using just the number line model why 3 5 21 35 starting with the number line de nition of 35 Explain using just the number line model why 3 5 21 35 starting with the number line de nition of 2135 4 Write the solution to the following division problem using a scaffold algorithm see the above chart for a reference If you use the version of the scaffold algorithm in which the numbers are written above the division line then you may choose the multiples of powers often as large as possible or you may include several guesses for the same power of ten For example in a division problem your scaffold of numbers may include the guess of 200 400 300 and these are multiples of 100 which is the power often 10A2 100 It would also be ok to use the version of the scaffold algorith where the numbers appear to the right of the division sign and guesses are made for the multiples of powers often 7463 The aim in this problem is to exhibit the scaffold algorithm with all its details Just computing the answer 746 3 248 R 2 is not the goal and on a test that answer by itself would earn no credit Also show how to express the answer as a mixed number 5 Write three different explanations of how one can describe the quotient 408 In the rst use the repeated subtraction model in the second the partition or sharing model and in the third use the missing factor model You do not need to include real life applications illustrating these models unless you feel including those helps to illustrate the mathematics Your answers should only be around two to four sentences each or less i Repeated subtraction model ii Partition or sharing model iii Missing factor model 6 How many factors does the number n 35 72 l33 l74 2184882549759 have How could you give a description of what those factors are in terms of the prime numbers 3 7 l3 and 17 appearing the in the prime factorization of n 7 Suppose two people one called A and the other called B are running a race They start at the same point on a circular track and have to run around the track a number of times If A can run all the way around in 92 seconds and B can run all the way around in 144 seconds how many seconds after they start will it be before they are both at the original starting point at the same time again 8 How many factors does 504 have 9 To check whether 9391 is a prime number what is the largest prime number which you would have to check to see if it is a factor of 9391 10 Find all threedigit numbers which are divisible by 5 and for which the sum of the digits is 18 11 Find all fourdigit numbers which are divisible by 5 by 11 for which the sum of the digits is 18 and which are less than 3000 Note this problem can be solved without using the 39divisibility by 1139 test discussed in the text section 51 But the solution can be obtained without too much work if you use the second theorem on page 190 the theorem closer to the bottom of the page concerning divisibility by a product ab 12 i Give an example of a story problem which uses GCF24 64 for its solution and solve the problem ii Give an example of a story problem which uses GCF24 64 76 for its solution and solve the problem 13 Give a reason or argument based on the number line de nition of fractions to show that 820 and 25 are the same starting with the number line de nition for 820 So your reason must begin with the number line de nition of 8 20 as a dot obtained in a certain way and show this 820 is the same as the dot obtained as 25 with the number line de nition Answers like you can cancel out the 4 s or cross multiply or they both equal 4 are not correct in this context because those do not use the number line de nition 14 Show the details for performing the subtraction problem 352 base eight 7 164 base eight USING THE EQUAL ADDENDS METHOD 15 Show the details for performing the addition problem 357 base eight 164 base eight 276 base eight USING THE SCRATCH MARK METHOD 16 Let x and y be natural numbers and consider the fraction xy Give a formula for the correct expression for xy in lowest form or simplest form using any combination of x y GCFxy andor LCMxy 17 For the number 90 nd two factors whose product is also a factor of 90 and nd two factors whose product in not a factor of 90 PRACTICE PROBLEM SOME SOLUTIONS 1 If a is a whole number give reasons why a 0 is not de ned as a whole number Your reasons should be like those in the text or like those discussed in class It is not enough to just so a0 is quotunde nedquot or quotnot de nedquot Your task is to explain WHY it cannot BE de ned as a whole number Solution to 1 The reasons are given in two cases a nonzer0 and a 0 If a is nonzero and you assume a0 is a whole number then you get an impossible conclusion that a is both nonzero and zero simultaneously Here is that argument suppose a is nonzero and suppose a 0 is a whole number Call this whole number c so a 0 c The missing factor model of division then implies that the equation a 0 c is the same as the equation a 0 c But 0 c 0 so we nd that a 0 c implies a 0 But we assumed ais NOT zero This is impossible The only way out of this impossible conclusion is to realize that our starting assumption that a 0 is a whole number is false Now assume a 0 and assume a 0 0 0 is a whole number c so we have the equation 0 0 c Again the missing factor model of division yields 0 0 c The problem with this equation is that it is valid for ALL whole numbers c That is c could be 01234any ofthese numbers and the equation 0 0 c would be true in every case So the expression 0 0 cannot be a whole number c because there is no uniquely determined whole number it must be One can give additional arguments to show that an expression like 0 0 can actually be made as close as you like to any real number this argument uses ideas called limits and the topic of limits is not part of what we will cover in 22M006 this semester 3 Give the number line model definition of a fraction a b with a and b positive integers Explain using just the number line model why 3 5 21 35 starting with the number line definition of 35 Explain using just the number line model why 3 5 21 35 starting with the number line definition of 2135 Solution to 1 The number line model definition of a fraction 61 b with a and b positive whole number is the following Take each one of the length1 intervals 0 1 1 2 2 3 and so on and divide up each one into bpieces ofequal length and mark each one of these pieces with say a small vertical line at its start and end Now the fraction a b means the dot on the number line obtained by starting at 0 and counting of a of these smaller pieces or count over a of the small vertical lines not including one at 0 To show 35 21 35 starting with 35 using the number line de nition we proceed in the analogous way The number 3 5 is obtained by dividing up each unit interval 0 1 1 2 2 3 etc into 5 pieces of equal length and marking each one ofthese pieces Then starting at 0 we count over 3 of these pieces and stop at the righthand point That point is the number line definition of 3 5 In pictures we start with the number line 1 1 1 a 0 l 2 Now divide up each unit interval 0 l l 2 etc into 5 pieces of equal length like this 1 quot1 19 1 2 0 15 25 35 45 55 65 75 85 95 105 Then 35 is the point on the line right under the big black dot The number 21 35 is obtained by dividing up each unit interval 0 l l 2 2 3 etc into 35 pieces of equal length and marking each one ofthese pieces Then starting at 0 we count over 21 of these pieces and stopping at the righthand point That point is the number line de nition of 21 35 In pictures we start with the number line 1 1 1 e 0 l 2 Now divide up each unit interval 0 l l 2 etc into 35 pieces of equal length in two steps Step one is to use the l 5 divisions we used in the rst part of the problem and step two is now further divide each one of these smaller l5sized pieces into seven pieces of equal length See the gures below Step 1 Step 2 Divide each one of these smaller pieces starting with this one l into seven pieces of equal length 0 l 5 The result looks like this for this starting interval l l and some of the remaining parts are 1lllll1H H 111 H H 1 0 0 15 25 35 45 55 65 What we now have is each unit length interval divided into 7 5 35 equal sized pieces Counting over 21 7 3 of these determines the same dot on the number line as counting over 3 ofthe 15sized pieces because each 15size piece contains 7 ofthe smallest pieces so 3 ofthe 15sized pieces counts 3 7 21 ofthe smallest sized 135 pieces This is a way to argue that 35 21 35 using only the number line de nition of fraction and not any other notion of fraction which gets introduced after the number line de nition of fraction Then 35 21 35 is the point on the line right under the big black dot We brie y explain why 3 5 21 35 starting with the number line de nition of 2135 start with 135sized pieces obtained by dividingeach unit interval into 35 equal sized pieces You count over 21 ofthose from 0 to locate the dote 2135 Now join together these l35sized pieces 7 consecutive pieces at atime These make up 5 groups in each unit interval Counting over 21 of the 135 pieces is the same as counting 3 of the ones joined together seven at a time bc 3721 You should be able to ll in more detail to this discussion and complete this argument 4 Write the solution to the following division problem using a scaffold algorithm see the above chart for a reference 746 3 The aim in this problem is to exhibit the scaffold algorithm with all its details Just computing the answer 746 3 248 R 2 is not the goal and on atest that answer by itself would earn no credit SOLUTION If you use the version of the scaffold algorithm in which the numbers are written above the division line the numbers appearing above the usual division line 17 may be just a scaffold or tower of three numbers 200 at the bottom 40 on top of that 8 on top of that then a line over the 8 to produce the sum 248 There might be other numbers in this tower if you included additional smaller guess for the powers of ten For example instead of 40 you might have included the numbers 20 10 and 10 in the tower If your scaffold or tower was had the three numbers 200 at the bottom 40 on top of that 8 on top of that then a line over the 8 to produce the sum 248 The numbers appearing below the usual division line l and below the 746 are 600 then subtraction line to 146 then 120 then subtraction line to 26 then 24 then subtraction line to the remainder 2 As a mixed number the answer would be written 248 2 3 5 Write three different explanations of how one can describe the quotient 408 In the first use the repeated subtraction model in the second the partition or sharing model and in the third use the missing factor model You do not need to include real life applications illustrating these models unless you feel including those helps to illustrate the mathematics Your answers should only be around two to four sentences each or less i Repeated subtraction model SOLUTION Many answers are possible One might be you have 40 cookies and want to make packages with 8 cookies in each package How many packages can you make This model is solved by computing the number of times a set with 8 items can be removed from a set with 40 items so it illustrates repeated subtraction ii Partition or sharing model SOLUTION Many answers are possible One might be you have 40 cookies and want to share them equally among 8 students How many cookies does each student get This model is solved by partitioning the set of 40 cookies into 8 subsets each with the same number of cookies 8 You determine eg by dealing out the cookies one at atime to each one of the 8 students that each subset ends up with 5 cookies So it illustrates the partition or sharing model iii Missing factor model SOLUTION more than 4 sentences are included bc the solution method is also described Many answers are possible one might be this Suppose you are a frog living in a pond with some lily pads in it If you make 8 big hops in a straight line starting from a lily pad and you end up 40 regularsized hops from that lily pad how many regularsized hops are there in a big hop This problem can be solved by considering how many regularsized hops there are in a big hop Calling the number of regularsized hops there are in a big hop the unknown X we want to find the value of X and what we know about X is that it satisfies the equation 40 8 X So X is a missing factor of 5 6 How many factors does the number n 35 72 l33 l74 2l84882549759 have How could you give a description of what those factors are in terms of the prime numbers 3 7 l3 and 17 appearing the in the prime factorization of n SOLUTION The 360 factors of n can be described systemically like this Every factor of n must have the form 321 7b13 17d where a b c and d are whole number which satisfy 0 S a S 5 there are 6 such values of a 0 S bS 2 there are 3 such values of b 0 S c S 3 there are 4 such values of c 0 S d S 4 there are 5 such values of a For example 31 72 130 173 is one factor of n as is 34 70 130 174 Taking all possible choices of a b c and din this way gives has 51213141 6345 360 factors 7 Suppose two people one called A and the other called B are running a race They start at the same point on a circular track and have to run around the track a number of times If A can run all the way around in 92 seconds and B can run all the way around in 144 seconds how many seconds after they start will it be before they are both at the original starting point at the same time again SOLUTION The solution is LCM92 144 seconds This is LCM92 144 seconds 3312 seconds Note that 92 22 23 and 144 24 32 so LCM92 144 24 32 23 3312 8 How many factors does 504 have Solution to 8 504 8 X 9 X 7 2A3 X 3A2 X 7 and this is the prime factorization Therefore the number offactorsis31X21X11 24 9 To check whether 9391 is a prime number what is the largest prime number which you would have to check to see if it is a factor of 9391 Solution to 9 939135 the square root of 9391 is appr0Ximately 969 Therefore the largest prime factor 9391 could have is the largest prime less than or equal to 96 This prime is 89 In order to answer this question it was not necessary to determine whether or not 9391 is prime It turns out that 9391 is a prime number although this in not obvious 10 Find all threedigit numbers which are divisible by 5 and for which the sum of the digits is 18 Solution to 10 Since it is divisible by 5 the digit in the ones place must be a 0 or a 5 If the digit in the ones place is a 0 then the other two digits must be 9 and 9 to make the sum 18 The other possibilities must all have a 5 in the ones place and have the other two digits add to 13 The only possibilities are 945 855 765 675 585 495 along with the original solution 990 and these are all solutions 11 Find all fourdigit numbers which are divisible by 5 by 11 for which the sum of the digits is 18 and which are less than 3000 Note this problem can be solved without using the 39divisibility by 1139 test discussed in the text section 51 But the solution can be obtained without too much work if you use the second theorem on page 190 the theorem closer to the bottom of the page concerning divisibility by a product ab Solution to 11 Because the sum of the digits is 18 the number must be divisible by 9 Therefore this number is divisible by 5 9 and 11 hence must be divisible by their product 5 X 9 X 11 495 this is a consequence of the divisibility by products theorem page 190 The multiples of 495 which give fourdigit numbers less than 3000 are 495 X 3 1485 495 X 4 1980 495 X 5 2475 495 X 6 2970 All four ofthese numbers are solutions

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