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# MATH FOR ELEM TEACHERS MA 201

UK

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This 12 page Class Notes was uploaded by Kennith Herman on Friday October 23, 2015. The Class Notes belongs to MA 201 at University of Kentucky taught by Staff in Fall. Since its upload, it has received 14 views. For similar materials see /class/228151/ma-201-university-of-kentucky in Mathematics (M) at University of Kentucky.

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Date Created: 10/23/15

Chapter 3 Notes for Instructors Content The third chapter of Long and DeTemple discusses place value and algorithms for whole number arithmetic I tested students on the material from the rst four sections I spent one class discussing material from 35 I think this is a fun topic This discussion was pretty relaxed so I only included one question from 35 on the exam I did not have time to cover the material in section 36 Moreover it is di cult to cover the calculator sections because we do not require that the students use a speci c type of calculator Notes and Suggestions Notes on Section 31Numemtz tm Systems Past and Present In section 31 Long and DeTemple introduce some of the numeration systems that have been used throughout history At rst glance it may appear that this section is extraneous and unnecessary but I have found that this section really does provide a foundation for the positional system It also introduces the concept of borrowing in subtraction and making exchanges with addition I tested students over the ideas from section 31 but I did not require that they memorize the notation from section 31 because I do not think the systems themselves are the point of section 31 Manipulatives Several types of manipulatives are discussed at the end of section 31 My personal favorites are the the classroom abacus and the base ten blocks We do not have a classroom abacus though you can improvise with some colored block and a sheet of paper We do have a set of base ten blocks The also mention Uni xTMcubes I do not believe that we have any of these but we do have multilink cubes which could be used in a similar mannar An Activity Note I have provided you with a worksheet that I had my students do in class I did not lecture on section 31 I found that it was better to have them work through the ideas of this section in small groups Of course I was willing to answer questions but they should be able to gure out a lot of the ideas from the reading I brought the Uni x block to class for the last quesiton on the worksheet I doubt the designer of these blocks intended them for this purpose but they did help the students Notes on Section 32N0ndecz39mal Positional Systems In section 32 students study the concept of place value in nondecimal system Nor mally students nd the nondecimal positional systems frustrating It is not uncommon to hear comments along the lines of I will never teach this stu to my students77 and Who would ever use this stuff77 With regards to the latter comment you can tell them that computer scientists use the base two and base sixteen systems With regards to the former comment7 you can tell them that you are trying to help them understand how their students will feel when they begin to learn about the decimal system Remember7 that the decimal system may be familiar to us7 but it will not be familiar to their students Moreover7 we want these future teachers to know more about place value than their students An Activity Note With Manipulatives They do not introduce place value cards here7 but I like these because they can be easily adjusted for problems in di erent bases and they are easy to make All you need is a piece of paper and some poker chips Long and DeTemple discuss place value cards in sections 33 and 347 but it does not hurt to mention them earlier Place value cards can be very helpful when converting between di erent bases For example7 suppose you want to nd the base six representation for fty four Start with a pile of fty four poker chips Place these in the ones place of a base six place value card When we are nished7 no place should have more than ve poker chips Remove a group of thirty six chips from the ones place and exchange it for one chip in the thirty sixes place This will leave eighteen chips in the ones place and one chip in the thirty sixes place Now remove a group of six chips from the ones place and replace it with one chip in the sixes place Do this two more times You will then have a place value card with one chip in the thirty sixes place and three chips in the sixes place Therefore the base six representation of fty four is 130m Notes on Sections 33 and 34Algorithms for Adding and Subtracting Whole Numbers and Algorithms for Multiplication and Division of Whole Numbers I spent several days on the material from sections 33 and 34 Students should be able to use manipulatives to justify algorithms for addition7 subtraction7 multiplication7 and division of whole numbers Moreover7 they need to understand that the traditional algorithms for computations with whole number are not the only valid algorithms I am particularly fond of problem 14 in section 33 l was surprised at the number of future teachers who would have told the hypothetical student in this problem that he or she had incorrect reasoning Many of the future teachers believed that the hypothetical student had only happened on the correct answer by luck An Activity Note With Manipulatives As I mentioned before7 I like to use place value cards because they are easy to make and I nd that students understand them Certainly you can use other manipulatives such as base ten blocks assuming that you are7 in fact7 working in base ten in a similar manner I am providing you with four examples of arithmetic problems which I have solved with place value cards to show you how you can use these cards with students Chapter 1 Notes for Instructors Content The rst chapter of Long and DeTemple provides a foundation for the rest of the book It is important that you cover this chapter especially if you plan to require your students to read the textbook I say this in part because Polya7s Problem Solving Principles are covered in Section 13 Every example problem in the textbook follows Polya7s problems solving methodology Although it is important to cover Chapter 1 it should be noted that it is not hard to imagine how one could spend months on the problems in Chapter 1 I devoted 3 weeks to chapter 1 but in retrospect I think it would have been better to spend only two weeks on this chapter There are many problems in Chapter 1 which could be used in an elementary classroom This helps to provide some motivation for the course There are also more sophisticated problems It should be noted that some students may nd the more sophisticated problems easier than those which appear to be more appropriate for an elementary classroom College students have been trained to use variables whenever possible Many problems can be solved without using variables For example elementary students will often attempt to solve problems by working examples Using a variable is one of the problem solving strategies discussed in Chapter 1 but there are many others which college students will need to recall Notes and Suggestions Since you will not have the luxury of spending months on Chapter 1 I want to highlight some important ideas and topics in Chapter 1 that will be useful to the students later in the course 0 Polya s Problem Solving Principles This approach is used to solve problems throughout the textbook It is particularly important that you highlight the Look Back77 step a step which is often ignored Pascal s Triangle This triangle is important because it can be used in the chapter on probability to evaluate combinations Gauss s Trick I like this trick I suppose that one could teach this course without discussing Gauss7s Trick but I have found problems later in the text for which this trick comes in handy Moreover I think there are some good ideas which come along with problems that require Gauss7s trick First Gauss7s trick uses a variable in a clever way Second students will be required to add equations This does appear to be a novel idea to some of them Finally I like to use Gauss7s trick to nd the sum of the rst 72 terms in an arithmetic progression because these problems have several layers Students must begin by nding a formula for the nth term of the arithmetic progression They must also understand how to get the formula for the n 7 1 term That is students can often understand why 1 2 100 100 99 1 but it is moredi iculttoseethat132n712n712n731 1 o Triangular Numbers This is another example where you can use Gauss7s trick Moreover there are several problems in later sections which use the triangular numbers 0 Work Backward It would be possible to teach the course without discussing this strategy at the beginning of the course If this strategy is needed later it could be discussed at that time Nevertheless I like this strategy I like to cover this strategy early in the course because you can use problems similar to Example 111 on page 50 to teach students how to write good solutions There are lots of variations on the game of NlM one example is described in Example 111 which you can include on homework assignments I also like these problems because I think they are easier to solve if you examine similar smaller problems that is if you begin with a smaller pile This employs another problem solving strategy Also students are likely to appreciate the value of manipulatives when solving these problems Many students nd that the problem is much easier to solve if they actually have coins or markers with which to work Chapter 2 Notes for Instructors Content The second chapter of Long and DeTemple is geared toward de ning the whole numbers and the arithmetic operations on the whole numbers The rst section provides an intro duction to set theory The second section focuses on equivalence cardinality and ordering the whole numbers The third and fourth sections de ne whole number addition subtrac tion multiplication and division These sections also introduce several models for these operations Notes and Suggestions Notes on Section 21 Sets and Operations on Sets 0 Because the de nition for addition is based on the union of disjoint sets it is important for students to have a basic understanding of some set theory In particular students will need to understand the following ideas 7 Union of sets Intersection of sets Disjoint sets Transitivity of inclusion Commutativity of union Associativity of union 7 Properties of the empty set i The inclusion exclusion principle Subset Proper subset You can do a lot of other things in this section but these are the essentials Students should be able to understand Venn diagrams and perhaps draw a few on their own but I do not know that this skill is necessary to understand the key ideas in the remainder of the chapter You will notice that l have placed a special emphasis on the union of sets I have done this because addition on the whole numbers is de ned using the union of disjoint sets Since the main objective of this chapter is to de ne arithmetic operations on the whole numbers the union of sets seems to be one of the most important operation on sets For example the commutative property of addition follows from the commutativity of union You will need to de ne subset and proper subset so that you can order the whole numbers These ideas can be slightly confusing for students but I think they are necessary for students to fully understand the relations 32 lt and gt If A C B students should be able to clearly explain why this is true That is they should argue that each element of A is also an element of B and that there is an element of B that is not an element of A Moreover they should understand that A Q B and B Q A implies that A B If time permits you should also de ne the complement of a set and the Cartesian product The latter is used as a model for multiplication and in fact can be used to provide an alternate de nition for multiplication The former could be used to formally de ne subtraction by the take away model Moreover students will need to be familiar with the complement of a set when they study the chapter on probability 0 You can make a lot of good TrueFalse questions from the material in section 21 These questions can teach students to read de nitions carefully You can also teach students how to construct good counterexamples for those statements which are false Moreover you will need to stress that you can show that a statement is false by providing a counterexample but you usually cannot show that a statement is true by providing an example An Activity Note When introducing the operations on sets I found that it was helpful to involve the students Speci cally I told the students with brown hair to stand on the right side of the room Then I told the students with brown eyes to stand on the left side of the room At this point they should see the need to have an intersection You can also talk about complements at this point because some students should not be in either set Moreover you can address the inclusion exclusion principle since it is likely that there will be students in the intersection of the sets You can certainly make this activity more elaborate if you like I do believe that the activity was for my class more interesting and e ective than the traditional lecture approach for teaching set theory Notes on Section 22 Sets Counting and the Whole Numbers 0 In this section Long and DeTemple de ne one to one correspondence equivalent sets the whole numbers and the ordering of the whole numbers 0 Manipulatives On pages 87789 Long and DeTemple discuss manipulatives that can be used to represent the whole numbers We do have the cubes discussed at the top of page 88 and Cuisenaire rods which are similar to the number strips shown on page 88 o Pacing In retrospect I do not think it was necessary to spend a whole lot of time on section 2 I also think you should skip the Hamming codes unless you really have a lot of extra time You will probably need quite a bit of time on sections 3 and 4 The Hamming code discussion at the end of section 2 appears to be an attempt to show the student that there are applications for set theory I do believe that applications are important7 but it is di cult to include them because of the pace of this course I found that it was more important to devote extra time to the central ideas of the course which students will actually be teaching themselves Notes on Sections 23 and 24 Addition and Subtraction of Whole Numbers and Multi plication and Division of Whole Numbers 0 ln sections 3 and 4 Long and Temple de ne addition7 subtraction7 multiplication and division of whole numbers It is important that students understand the di erent models of arithmetic discussed in these sections Moreover7 it is important that you de ne the arithmetic operations as they have done in the book because these de nitions generalize easily 0 Addition I like to discuss the importance of the word disjoint in the de nition for addition This can be done by providing an example An Activity Note Sometimes it is useful to involve the class Determine how many students have brown hair and how many students have green eyes Ask them to determine how many students have brown hair or green eyes If you are lucky7 your sets will not be disjoint Ask them how the problem would be di erent if we wanted to determine the number of students who have green eyes or blue eyes Students should use the set model when combining two groups and the number line model when looking at distances The properties on page 101 follow easily from the de nition of whole number addition and the properties of sets given on page 78 o Subtraction Students may think it is strange to de ne whole number subtraction by the missing addend model I nd that it is useful to remind them that they rst learned division by the missing factor model If students want to de ne whole number subtraction by the take away model7 I ask them how they would de ne subtraction for the integers It is di cult to think of taking away a negative number We use the missing addend model to de ne subtraction because it generalizes easily to other number systems We are looking ahead toward the big picture 0 Multiplication The easiest model for students appears to be the repeated addition model That is familiar to them i I think the multiplication tree model is di cult to understand because they do not put labels on the tree When I labeled the nodes and the nal outcome at each leaf of the tree they seemed to understand it better Multiplication trees will show up again in the second semester when they study probability 7 The Cartesian product model will show up again when students study probability in the second semester 7 The rectangular area model for multiplication is IMPORTANT It is the only model given in this section for whole number multiplication which can be adjusted for rational number multiplication I really like the way that the textbook uses the area model to explain the foil method to students 0 Division 7 The partition model for division tends to be the easiest model for students to un derstand though they will need to understand all three models given for division 7 Many students are unaware that division is repeated subtraction It is therefore not surprising that some of them will not remember how to do a long division problem without using a calculator For them the long division algorithm is simply an algorithm without meaning The long division algorithm should ow naturally from this model i The missing factor model is used to de ne division because as with the missing addend model for subtraction this de nition will generalize easily to other number systems i It is a personal pet peeve of mine that students do not know that you can divide zero by a nonzero number but you cannot divide any number by zero I believe that this is because they do not fully understand division Certainly they can memorize this fact for the exam but I think it is important that they be able to explain why this is the case using all three of the models for division I have attached a sheet with my explanation for the three models I do require that they provide these types of explanations on exams Chapter 7 Notes for Instructors Content The focus of this chapter is the Real Number System Up to this point in the course the need for richer number systems has been obvious because we desire to have closure under the basic operations of arithmetic For example in moving from the Whole Number System to the lntegers we gain closure under subtraction It may not be so obvious why we need to move from the Rational Number System to the Real Number System Consequently I think it is important to prove that the xi is irrational Beyond that I did not have much time to spend on Chapter 7 Consequently I spent very little time on the last two sections of this chapter it was unfortunate but necessary Manipulatives We have the base ten blocks which can be used to represent decimals Note that it is important that you do not call the small cube the unit when working with decimals The unit will have to be the long the at or the large block depending on the magnitude of the numbers you wish to represent For example if I needed to represent 1345 then my unit would be the long and I would represent this with one large block three ats four longs units and ve small cubes On the other hand if I wanted to represent 2356 then my unit would be the at and I would represent this with two large block three ats units ve longs and six small cubes You can also use money to represent decimals I havent seen any problems with this approach but there does appear to be some discussion about this technique among educators l7m not sure what their objections are but you should be aware that a certain faction of educators is somewhat leery of this approach The place value cards that we used to represent whole number arithmetic can be easily adjusted to represent decimal addition and subtraction Notes and Suggestions Notes on Section 71 Decimals o I believe that it is important to prove that the xi is irrational This proof will provide students with a wealth of irrational numbers In order to understand this proof stu dents will need to be reminded about the Fundamental Theorem of Arithmetic You will also need to familiarize them with the following idea based on the FTA If a and p are integers p is prime and p is a factor of a2 then p is a factor of a You should also convince students that this is not true if we remove the restriction that p is prime I wanted my students to be able to do problems similar to those in Examples 73 74 75 and 76 on pages 4197421 of the textbook Notes on Section 72 Computations with Decimals c When working with terminating decimals7 students should see the connection between fraction arithmetic and decimal arithmetic 0 ln theory7 these students should see scienti c notation and signi cant digits in a science class7 so I tried not to spend too much time on this section Notes on Sections 73 and 74Ratio and Proportion and Percents o I spent very little time on these sections I did try to ensure that they could do basic calculations with percents similar to those in problems 1711 of section 74 Any extra time you have to devote to these sections would certainly be worthwhile Worksheets l have included two worksheets with this documentation Chapter 5 Notes for Instructors Content The integers are the focus of this chapter I only covered sections 51753 of this chapter Because I spent a lot of time on the properties of whole number arithmetic in chapter 2 the properties of integer were fairly evident to the students I did point out the new properties that we obtain when we go from whole number arithmetic to integer arithmetic but overall I spent very little time on the properties of integer arithmetic For example I mentioned that the whole numbers are not closed under subtraction but the integers are closed under subtraction Manipulatives Colored counters are the only manipulatives discussed extensively in Chapter 5 We do not have red and black colored counters but we do have orange and yellow colored counters in the Mathskellar I prefer the red an black counters because of their connection to the monetary phrases in the red77 and in the black77 but the students did not seem to mind the orange an yellow counters We also have multicolored centimeter cubes in the Mathskellar If you want to take a few minutes to separate the red and black cubes from the other colors you could also use these for the colored counters Notes and Suggestions Notes on Section 51Representatz390ns 0f Integers o It is important that students understand that there are many ways to represent an integer with colored counters They will need to understand this if they are going to use take away to demonstrate subtraction problems 0 Students often have dif culty with the de nition for absolute value They know that absolute values should be positive but they fail to understand that 7a is positive when a is negative Notes on Section 52Addit2390n and Subtraction of Integers o The take away model for subtraction of integers is considerably more dif cult for stu dents than the take away model for subtraction of whole numbers When we use colored counters and the take away model to represent a 7 b we will need to be cer tain that the colored counter representation we use for 1 contains a colored counter representation for b For example if I want to evaluate 5 7 73 using take away and colored counters I will need to have a colored counter representation for 5 which con tains a colored counter representation for 73 so that I can take away 73 So ifI use ve black colored counters to represent 5 then I cannot take away a representation for 73 On the other hand if I use for example ten black counters and ve red counters to represent 5 then I can take away three red counters which represents subtracting 73 This will leave me with ten black counters and two red counters This group of colored counters represents 8 Hence 5 7 73 8 o I think the di iculty students have With the take away model for integer subtraction provides motivation for de ning subtraction using the missing addend model The take away model Which is easy to understand for Whole number subtraction does not generalize easily to subtraction With other number systems 0 Many of my students had di iculty With the number line model that they use for subtraction lt is7 in essence7 illustrating the missing addend model for subtraction Notes on Section 53Multiplicati0n and Division of Integers o I do like the justi cation for integer multiplication in the textbook that uses patterns I actually had my students work through a similar exercise in class 0 I really like to assign question 8 in section 53 for homework I found that some students needed practice in identifying the properties of integer arithmetic

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