difficult for the average American elementary school teacher. ...... assume that I am preparing a shipment of 4 pallets, 2 tiers, 3 six-packs, and 2 bottles for Cub .... But then the woman becomes concerned over the cost of reassembling the necklace at .... the bellboy $5.00 and tells him to give it to the men. ... missing $1.00? 3.
What you have before you are is an unfinished ongoing arrangement of a summary of some notes—apportioned by chapter—that I am currently accumulating and have accumulated over time. These notes are written so that they might be easily incorporated in a subsequent revision of the book Number Theory for Elementary School Teachers; however that is unlikely to ever occur. Use them as you wish.
Please note that these notes are yet incomplete. I will rectify that as I find the time. An indication of my progress is the version number of these comments. When I reach 1.0 they will be relatively complete.
Preface: Comments This book Number Theory for Elementary School Teachers was an outgrowth of a workshop I offered to New York City Mathematics Teachers that began in the summer of 2007 and a mathematics methods course I taught for a number of years for certified elementary school teachers returning to take course work for their Masters degree. However, the inspiration for the book came from several pre-service elementary school Masters students who, over the years, voiced concerns that there were no suitable mathematics courses beyond those usually required for pre-service elementary school teachers who wished to further explore the mathematics of the elementary grades. Consequently, when in the fall of 2007 I was offered the opportunity by the then Dean of the School of Education—Alfred Posamentier—I immediately set to work preparing this volume. I had, over my years of teaching mathematics and observing the teaching of mathematics, come to the conclusion that much of the elementary school mathematics curriculum was, in a sense, number theoretic in nature and, hence, my choice of the content and title you see before you. I attempted to write the book so that it could be read outside a formal setting—this was one of the reasons1 I embedded problems within the text and hints within the problems at the end of each chapter—however, I also intend that it be used in my own teaching and that of others. 1
Another reason is that I wished, in a sense, to make this book writeable (to gain a sense
2 Unfortunately, writing this book for two separate audiences has weakened its appeal and certain of the chapter problems while reasonably appealing to middle school teachers seem somewhat difficult for the average American elementary school teacher. In consequence, over the years, I have developed an alternate set of problems to use with elementary school teachers. Quite frankly, I considered this book an experiment that might be bettered by others; something that has regrettably not yet happened or of which I am unaware. Between 2007 and 2014 I used this book or portions thereof in workshops and mathematics methods courses. In the mathematics methods course I alternated a week from the book with a week of discussion around pedagogy and seldom progressed beyond the chapter of on rational numbers. However, in the years between 2011 and 2014—essentially five semesters— I was asked to give a series of semester workshops for elementary school teachers that focused exclusively on the mathematics of the elementary school grades: Whole Number Arithmetic Real Number Arithmetic Algebra and Functions Geometry and Measurement Probability and Statistics (1/2 semester) The final 1/2 semester was a practicum where I and this cohort focused on implementation of what they had learned. I have reflected on this final 1/2 semester in an essay which I may post in the future. The book was used the first two semesters. I have my notes for the last three semesters and may, at some point in tine, put them in some sort of coherent order for dissemination.
3 Mathematical Explanations and Arguments: Comments I put this chapter first as I felt it was foundational. In my observations of elementary school classrooms I see children in or on the verge of apodeictic proof more than often unaccompanied by an corresponding awareness in their teacher. Unfortunately, and I admit to a bias, this means that many children miss out on much of the taste and beauty of mathematics. Postulational Proof Early on in my teaching of elementary school mathematics methods I realized that many of the teachers and teacher candidates with whom I worked had considerable skills in language arts. Attempting to build on these skills I put together a lesson which develops2, using word ladders, a language arts analogue of postulational proof, and enacted this lesson in connexion with material in this chapter—or some suitable modification thereof. Proof Lesson Word Ladders Open Ladder Write the word PARK on the board and tell the students that they are to make as many four letter words as they can within 15 minute period. However, they must begin with the word PARK and they can only change one letter at a time, each time obtaining a proper English word (that is, one that is in a usual English dictionary). For example PARK -> PART; PARK -> PART -> PAST 2
I became aware of the possibility of using word ladders as an analogue while browsing the Internet—I have long forgotten the page—however, something similar may be seen in Ian Stewart’s Nature’s Numbers and there perhaps a degree of cross-pollination at http://mathforum.org/sanders/exploringandwritinggeometry/extracredit.htm
4 or PARK -> MARK; PARK -> MARK -> LARK When the allotted time is up have students share their solutions and their strategies. Some strategies that might be voiced are ▪
Held first and last letter constant and changed the middle letters
▪
Tried the letters in order
▪
Made a new word and then looked for another new word to make
▪
Blends
▪
Word Families ; e.g. get an 'e' at the end It is useful to ask why a particular strategy was chosen. Make the point (1) that while some people may have proceeded somewhat randomly,
others made their decision based on their knowledge of3 and experience4 with words and (2) children have similar experiences in their exploring of number in the elementary grades. That is, if a child has little pragmatic knowledge of5 and experiences with number in comparison with other classmates, she may find it difficult to respond effectively to mathematics assignments or homework. Closed Ladders Tell students that you are going to begin with the conjecture DOG -> CAT
3
I have in mind here word structure. It was often the case that a number of my students had learned English as a second language, and this was usually evidenced in the shortness of their list. This lack of knowledge of and experience with English has the potential of significantly reducing the effectiveness of this sort of lesson. 5 I have in mind here number structure. 4
5 that is, you hypothesize that DOG can be changed into CAT by changing one letter at a time and at each change obtaining a proper English word. Make the point that such conjectures may not be true and that the only way to determine whether they are true is give a proof; that is, to demonstrate that there is an appropriate chain leading, for instance, DOG to CAT. Give students a few minutes to work on this problem and then ask for proofs. A proof, by the way, might be DOG -> DOT; DOT -> COT; COT -> CAT; As previously ask for strategies and reasons for using such. Then consider the conjecture that DOG -> RABBIT Ask students what they think and why they think that way. Ask them if they can prove this conjecture is false. Then ask them about this conjecture SHIP-> DOCK Give them some time to work on this conjecture individually and with partners if they already aren't so doing. Summing Up Develop a baseline for discussing problem solving using their experience with word ladders. This is a critical juncture! Develop an agreed upon list of strategies and list some the ways the students handled getting stuck; revisions; cooperation and respect; flexibility; holding on; play; and elegance. That is, were there particular solutions to the conjecture that the students thought were more elegant than others? For instance, often and as with mathematical proofs shortness will be seen as a marker of elegance.
6 All classes I have taught enjoy the word ladder exercise and, depending on the class, I often assign for optional homework the conjecture BLOOD -> STONE There is a very elegant solution at http://archive.hbook.com/magazine/articles/2000/sep00_armstrong.asp Word Problems Tell the students that they are going to use some of the strategies they used with word ladders on some mathematical problems involving what mathematicians call proof and define, informally, proof as an argument that begins with agreed upon assumptions—for instance and with word ladders, one begins with a word ‘CAT’—a goal—for instance and as with word ladders, one has a word that one wishes to transition towards, ‘DOG’—and proceeds step by step in a logical fashion—for instance and as with word ladders, CAT -> COT; COT -> DOT; DOT > DOG. In illustration, have students work on the following problem while noting, at the outset, that ▪
the sum of two even numbers is even
▪
The sum of two odd numbers is even
▪
The sum of an even and an odd number is odd
and checking to make sure all students are comfortable with these facts. Then conjecture that The sum of two consecutive counting6 numbers is never divisible by 4 Give a few examples such as 10 + 11 and 21 + 22, but emphasize that students need to either prove or disprove this conjecture for any two consecutive counting numbers. 6
It may be necessary to make this clear and often I use ‘natural’ in place of ‘counting’ anticipating a subsequent chapter in the book.
7 A number of students usually are able to provide a reasonable proof. Push a bit for proofs that are least algebraic, but highly logical; it may be necessary as a class to fill in the blanks. However, accept all and be sure to take up the frustrations of those that get stuck. Continue to point back to the work with word ladders! Give as homework the following conjectures 1.7 The sum of two consecutive counting numbers is never divisible by 4 2.8 The sum of three consecutive counting numbers is never divisible by 4 39. The sum of four consecutive counting numbers is never divisible by 4 Over the years I have speculated as to whether there might be another series of problems that would as effective for introducing proof as the above, but would be more accessible or, in a sense, more relevant. At the next class session review the homework emphasizing strategies and insights gained from the work on word ladders. Also emphasize the need, in the case of these particular problems, to take into account all the counting numbers. Point out that by third grade we, in affect, ask children to do something similar when we, as teachers, tell them, “An even number plus an even number is always even10.”
7
As some students nod their head in understanding even though they may be unsure, I ask again for the first of these conjectures to see if they can, at least, reproduce the argument(s) that was given in class. 8 This conjecture happens not to be always true. A number of students consider only a few cases—usually those for which it is true—and conclude otherwise. This makes the point that they must, in some fashion, consider all cases. Sometimes a student will be able to characterize when the conjecture is false and why. This seems to be worth noting in any case as it makes the point that there is often something further to be discovered. 9 This has a nice non-algebraic proof. Consider as an illustration: 11 12 13 14. Take 1 from the 14 (the fourth number) and add it to the 11 (the first number); i.e. 12 12 13 13. Adding these four numbers you have 2 (12 + 13). However 12 is even and 13 is odd so their sum is not divisible by 2. Hence 2 times 12 + 13 is not divisible by 4 and , hence, the sum of the four numbers is not divisible by 4. 10 Third graders are quite capable, by the way, of giving a convincing proof that this is indeed so.
8 Proofs by Contradiction Although I do give a proof by contradiction in the first chapter, I did not spend much classroom time developing the idea. One reason for this omission is that while one assumes the conjecture is false, the following contradiction arises through a series of what might be termed apodeictic steps. Hence so I thought, “if students understand postulational proofs then proofs by contradiction will be somewhat self-evident.” This was not the case and in retrospect something like the following might have been more helpful than what I included in Chapter 1: Number Go On Forever Let n be any natural number. Assume that n +1 is a natural number. Prove there is no largest natural number. Proof: Assume there is a largest natural number N. However, N + 1 is a natural number and N + 1 is greater than N. This is a contradiction so there must be no largest number. Note that this proof simply formalizes what children seem to mean when they say, “Numbers go on forever.” Note that this proof simply formalizes what children seem to mean when they say, “Numbers go on forever.”
Proofs by Induction Similarly I give a proof by induction in the first chapter and again I did not spend much classroom time developing the idea. This omission is, perhaps, more serious since such proofs seem to naturally arise from patterns children observe on the number line. Hence, in retrospect
9 the following might have more a more appropriate beginning example or, at the least, might have been used as a follow-up: Closure over Addition Assume for any natural number M that M + 1 is a natural number. Let n and m be any two natural numbers then n + m is a natural number
(A)
Proof: Well, (A) is certainly true if m = 1. So assume it is true for m = N; that is n+ N is a natural number How about m = N + 1? Well, we know11 that (n + N) is a natural number and since numbers go on forever, (n+N) + 1 is a natural number. Thus by the associative law of addition n + (N + 1) is a natural number as was to be shown.
Final Comments In subsequent chapters of the book I revisit proof and provide problems that require proof; however, I have found this does not necessarily produce—especially for many elementary teachers—a substantial awareness of apodeictic proof (and, in effect, proofs by contradiction or 11
Remember my original assumption
10 induction). I will try to rectify this lack in subsequent chapter commentary by (1) sketching short proofs that demonstrate that certain arithmetic facts are not as self-evident as they might seem, and (2) designating certain problems for proof that I feel are relevant to the chapter at hand and that might presumably be discussed by the class and teacher in terms of their logical structure; problems that are, one might say, writeable12. I regret to say that I have not yet had the opportunity to substantially test such an intervention. Much of my focus in this chapter is on the mathematics done by children in prekindergarten, kindergarten and the early primary grades. I begin, after a brief sketch of ancient numbers systems, with thinking about is entailed in counting and then move to sections on positional number systems and large numbers. Over time the section on counting has evolved in my thinking—perhaps more so than other sections.
12
With the term writeable I am attempting to point towards a type of problem whose solution can uniquely be re-produced and not merely understood.
11 The Art of Counting: Comments Much of my focus in this chapter is on the mathematics
done by children in prekindergarten,
kindergarden and the early primary grades. I begin, after a brief sketch of ancient numbers systems, with thinking about is entailed in counting and then move to sections on positional number systems and large
numbers. Over
time the section on counting has evolved in my thinking—perhaps more so than other sections. Counting in some reasonable precise manner requires that (1) the items to be counted be arranged in some accessible fashion and (2) successive ordinal numbers (usually beginning at 1) be assigned uniquely to each item. Sets Although my focus in the book is on number, young children spend considerable time exploring the properties—for instance, color, size, and weight—of various objects and devising schemes for their groupings. In hindsight I should have referred to class discussions at this juncture when I took, for instance, geometry. Formally, I observed three set operations: •
Union (or exclusive ‘or’)
•
Intersection (or ‘and’)
•
Not13 ( or ‘other than’) As you can imagine such formal notions of set were a bit intimidating and, thus, we, as a
class, engaged in more pragmatic work. Teachers found both Venn diagrams and certain set classification puzzles developed by SET Enterprises, Inc. intriguing. I note14 in this later case that SET Enterprises, Inc. has developed applications for both iPhone and iPad (even a daily puzzle). Again, in hindsight, I should have played such games—with a slightly different 13
This doesn’t translate well into common language. Say, for example, I was counting girls in a classroom then the boys in the classroom would be, for example, not ‘girls’ as regards students in the classroom. 14 Layman Allen developed in 1965 a board game called On Sets which is one of several games played in academic settings. There are also a series of logic games developed by Michel and Robert Lyon of which Meta-Forms and Logix are two.
12 emphasis depending on the area of mathematics to be discussed—with teachers at the beginning of those aforementioned workshops. In light of all this, I made drastic changes to the problem sets for this chapter. For instance, the following were used as replacements: 1. A town board consists of six people: Al, Bo, Cy, Di, Ed, and Flo. To be approved, a resolution must receive the votes of a majority (more than half of the board members. List all possible successful voting coalitions as subsets of the set of town board members. 2. Let M be all the months of the year; let A be the set of months that have exactly 30 days; let B be the set of months that begin with the letter “J”; let C be the set of months that contain the letter “r.” Describe the following sets: a. a.
A∪B
b. b.
B∩C
c. c.
M-C
d. d.
(M – B) ∩ C
e. e.
B ∪ (M – C)
3. Yolanda asked 100 coffee drinkers whether they like cream or sugar in their coffee. According to the Venn diagram below, how many like a. Cream? b. Sugar? c. Sugar but not cream? d. Cream but not sugar? e. Cream and sugar? f. Cream or sugar?
13 4.
A restaurant menu offers three choices of ice cream: Vanilla, Chocolate, and
Strawberry. One night there were 97 customers who ordered scoops of ice cream. There were 60 orders that included scoops of vanilla ice cream, 32 orders that included scoops of vanilla and chocolate, 20 orders that had scoops of all three flavors, and 28 orders that included scoops of vanilla and strawberry. (Hint: Draw a Venn diagram to represent the situation.) a. How many orders included only scoops of vanilla? b. In addition there were 55 orders that included scoops of chocolate. i.
How many orders included scoops of strawberry only
ii.
If there were 7 orders that included scoops of chocolate and strawberry only, how many orders included only scoops of chocolate?
Functions The language of functions reasonably describes what students do when they count. However, I soon found that the formalization that I used in the book was somewhat esoteric for teachers. It turns out, perhaps not surprisingly, that they were already using in the primarily grades an analog; that is, the function machine:
For instance, the function f(x) = x + 1 mapping the natural numbers into the natural numbers would be diagramed as
14
with the understanding that inputs and outputs fall within the natural numbers; that is, the domain of f corresponds to the inputs—here the natural numbers—the co-domain of f corresponds to the possible outputs—here the natural numbers—and the range of f corresponds to the actual outputs—here the natural numbers less 1. Note that if the function machine with input set {1, 2, 3) has the rule In
Out
1 -> 2 2 -> doesn’t given an output 3 -> 4 or In
Out
1 -> 2 2 -> 3 3 -> 4 and 515 then, by usual classroom conventions, the function machine must be broken and, in fact mathematically, neither of these mappings are functions. Hence, in a certain pragmatic sense, the definition of function and the characterization of a well-behaved16 (not broken) function machine
15 16
For instance, when inputting ‘3’, you get 4 or 5 randomly as output. I’m not sure what a good label might be
15 are analogues. This allowed me to rephrase several of the end of chapter problems I originally assigned: 1. Which of the following17 rules characterizes an well-behaved function machine: a. {(1,1), (2,1), (3,1), (4,1), (5,1)}
co-inputs18 = {1, 2,3, 4, 5}
b. {(2,1), (2,2), (2,3), (2,4), (2,5)}
co-inputs = {1, 2, 3, 4, 5}
c. {(0,2), (0,3), (0,4), (0,5), (0,6)}
co-inputs = {1, 2, 3, 4, 5}
2. Given the function machine with rule A = {(5,2), (7,4), (9,10), (x, 5)} and co-inputs = {2,4,10,5,34}. What value of x will make this a well-behaved function machine? 3. Let the input of a well-behaved function machine with rule ‘5x + 1’ (that is, 5 times an input + 1) be all the whole numbers and let its co-inputs be the same. What do you get when you input ‘5’, ’25’? If f is a function with domain A and co-domain, then f is said to be a one-to-one function if its inverse g—that is, the map taking elements of its range back to the originating elements of its domain—is a function with domain B and co-domain A. In the language of function machines, a well behaved function machine F with rule R, inputs A and co-inputs B is a oneto-one function machine if the function machine G characterized the the inverse of rule R, inputs B, and co-inputs B is a well-behaved function machine. In example, consider the following well-behaved function machine F
17
I have used the ordered pair notation to represent rules. It is possible that this is too great a leap at this time. In any case, (a,b) is defined as In a -> 18
Out b
I am using the awkward terminology co-inputs to designate the set of possible outputs.
16
with both inputs and co-inputs the natural numbers. The inverse function machine G is
that is, In
Out
2
->
1
3
->
2
and so on However, In 1
->
Out 0
17 and ‘0’ is not a natural number. Thus the well-behaved function machine F s not a well-behaved one-to-one function machine. It is important to note the impact that inputs and co-inputs have on well-behavior. If I had limited the co-inputs of F to {1, 3, 4, ….} the F would not have been a well-behaved function machine. In a similar fashion, if I had limited the co-input of F to all natural numbers with the exception of 1, then F would have been a well-behaved one-to-one function machine. Finally, if I had specified the inputs and co-inputs to F to be the integers, then F would have been a well-behaved one-to-one function machine. In the light of these definitions I can rephrase problem 2.4 in the book Let S = {1, 2, 3, 4, 5} and C = {1, 2, 4, 6, 8, 11} and let F be the function machine with inputs S and co-inputs C using the rule ‘2x’ (or 2 times) a.
Is F a well-behaved function machine? Why or why not?
b.
Let P = S - {5}. Let F—same rule as above—have inputs P and co-inputs C. Is F a well-behaved function machine? Why or why not?
c.
Let Q = C - {11}. Let F—same rule as above—have inputs P and co-inputs Q. Is F a well-behaved function machine? Why or why not? Is F a well-behaved one-to-one function machine? Why or why not?
and the end of chapter problems I originally assigned involving one-to-one functions: 1.
Let the input of a function machine F with rule ‘5x +1’ (that is, 5 times an input + 1) be all the whole numbers and let its co-input be the same. Then is this a wellbehaved function machine? Why or why not? Is it a well-behaved one-to-one function machine? Why or why not?
2.
Let the input of a function machine F with rule ‘0’ (that is, all inputs map to 0) be all numbers and let its co-input be the same. Then is F an well-behaved function machine? Why or why not? Is it a well-behaved one-to-one function machine? Why or why not?
18 Inequalities At the time I wrote this book, I was not sufficiently aware of the difficulties teachers had with inequalities. Among my sensitizing experiences two stand out. ▪
Among the problems I assigned in conjunction with the material in the first chapter was the following: Lemma I Let a, b, and c be natural numbers and assume that aB. In the very same situation, one is less than the other, mathematically B, B A-X= B+X A’=B’ B’=A’ Then of course there’s more fun when Katya’s in on it and transitivity pops in so that even without direct perceptual comparisons mathematics comes to the rescue so you can figure out stuff you wouldn’t know otherwise (do I smell motivation here?): A>B B>K A?K A>K and they work out proudly that you keep the ? (don’t know symbol) answer in the following situation A>B A>K B?K It remains forever a ? for mathematics, maybe direct perception will help but the current mathematics for the current situation takes a pass on it. We might use mathematics to come up with some nice questions and suppositions and come to more or less likely answers but…And then you can get to precision with measurement tools that work for the kinds of objects and …
21 Intriguingly the possibility of an alternate24 elementary mathematics curriculum was also explored25 by Louis P. Bénézet in Manchester, N. H. In 1936 where he, in effect and as principal, delayed instruction in formal arithmetic until sixth grade. Nonetheless, there was much arithmetic taught in grades 1 through 5. Carl. L. Thiele, who visited the Manchester, N.H., notes that Bénézet provided conclusive evidence that children profit greatly from an organized arithmetic program which stresses number concepts, relations, and meaning.” Guy. T. Buswell found that Bénézet had only deferred formal arithmetic, and that all other aspects of a desirable arithmetic curriculum were present. Leo. J. Brueckner says, "From these studies the conclusion should be drawn not that arithmetic should be postponed, but that the introduction of social arithmetic in the first few grades does not result in any loss in efficiency when the formal computational aspect of the work is introduced later on, say in grade three." I find myself wondering about similarities between what Bénézet and Davydov proposed and accomplished. Is there, for instance, a developing awareness of mathematical structure when children are engaged in what has been termed natural arithmetic? Combinatorics I yet unsure if including, at this point, a brief introduction to combinatorics was appropriate; however, I do draw on an acquaintance with combinatorics in Chapter 5 so perhaps I need to take it up again there in more detail.
Positional Number Systems The key idea in this chapter of the book is that, for example, any n+1-digit number in base-10 can be written26 as
24
See also Free At Last by Daniel Greenberg These quotes are taken from What does Research say about Arithmetic? By Vincent J. Glennon and C. W. Hunnicutt, National Education Association, Washington D. C., 1952, page 17-18. 26 This is termed its base-10 expansion 25
22
where 9 ≥ (i =0,..,n) ≥ 0. In the early grades one can see teachers working to instill such notions (using terms such as tens or hundreds, of course) and the very utterance of numerals is a sort of reinforcement. For example the base-10 number 324 is said “Three hundred, twentyfour”; that is, “three times one hundred plus three times ten plus four.” Counting This chapter, in effect, found its way into every mathematics methods courses I taught. While I have never met an elementary school teacher candidate who could not count above one hundred or who was not fairly adept in multi digit addition and subtraction, few of those understood the mathematics behind their success or fully appreciated how difficult an initial exposure to all this was for many of their own students. I normally began with discussion by constructing on the blackboard two columns. In one I wrote “-“ and then the letters A through F and in the other I drew blocks. In the row occupied by “-“ I drew no blocks. In the row with A I drew one blocks; B two blocks; and so on to F where I drew 6 blocks. I then told the teacher candidates that these were the first six digits in a place-value numbers system much like our base-10 system (at times I even added that this was a base-6 number system) and there were only six single digits. I then asked them, in that case, what came after F. Often a student will say reasonably seriously ‘G’; however I reply there are only these six single digits and note that in base-ten system the single digits are restricted to 0 and 1 through 9. I then point out in our base-ten system 10 comes after 9. Often somebody shouts out A(said “A dash”). I then ask them what comes after AF and many say B- and then I ask what comes after FF. Usually with only a little prodding, many say A—. We then spend a little time as a class if converting from the is representation of base-6 to base-ten with my emphasizing that a place in base-6 has the value of a power of 6 just as in base-10 it had the value of a power of 10 so that CD is 3*6 + 4 (i.e. 24). At this point I simplify the notation and, for instance, we evaluate, as a class, 346. I normally assign for homework a problem set such as
23 1.
Convert (a) 1216 to base-ten; (b) 1213 to base-ten; (c) 12116 to base-ten
The following class period I, using popsicle sticks and bundling, usually discuss how multi-digit counting is often taught in the primary grades. I began by counting, for example, 112 sticks by bundling into tens—that is 100 sticks (or 10 tens), 20 sticks (or 2 tens), and 1 stick (no tens)—and then, after giving my students a little practice, do the same counting in, for example, base 6. For example, 112 counted in base 6 would consist of 3 bundles of 36 (that is, 62), 0 bundles of 6, and 4; that is 3046. I normally assign for homework a problem set such as 2.
Convert (a) 17210 to base-5; (b) 17210 to base-2; (c) 17210 to base-1627
Some years I have assigned a problem such as 3.
If I have 112 sticks, then it is clear, as we have discussed, this can be written as
11210. However, I could have counted by ones or skip counted by twos to reach this number so I how do I know this representation is unique. That is I might be able record, using the usual notation of base-ten, this pile of sticks in two different ways . Prove that this cannot be the case; that the representation of any number in base-ten is unique. [Hint: A reasonable way to go about this is a proof by contradiction. That is, assume 112 = abc— where a, b, c are the usual base ten digits—and show you get a contradiction if a is not 1, b is not 1, or c is not 2). However many of the teacher candidates find this problem quite challenging. I do know of teachers that, so as to impress on their students the structure of the base-ten notation, work with their primary school students in different bases. I do not know, however, the effectiveness of such instruction. On the other hand, such work is a component of Davydov’s first grade mathematics curriculum.
27
Most mathematics educators that do something somewhat similar tell me using base-16, at some points, is essential to really cementing the idea of positional notation in their students’ minds.
24 I have seen other mathematics educators make such work in other bases a little more hands-on. For example, bottled water is sold by the bottle or by the six-pack. Imagine that such bottles of water come from the warehouse stacked on a pallet in six tiers each consisting of six six-pack. Then we have the following notation Single Bottles Packs (1 pack = 6 bottles) Tiers (1 tier = 6 six-packs) Pallets (1 pallet = 6 tiers) In such an environment it is not unreasonablet to imagine one person saying to another, “They want 4 pallets, 2 tiers, 3 six-packs, 2 bottles down at Cub Foods” when informed by the dispatcher that 956 bottles have been ordered. All this raises some interesting mathematical questions. It should be fairly evident that, given an order of bottles, it is always possible to appropriately arrange this in bottles, packs, tiers, and pallets so as to satisfy. However, in real life there are probably more or less than 6 packs to a tier and probably more or less than six tiers to a pallet. Thus one could wonder (1) if a tier holds 7 packs and a pallet holds 10 tiers, how many ways might an order of 456 bottles be filled and (2) if teach tier holds 6 packs and each pallet holds 6 tiers, how many ways might an order of 456 bottles be filled. The answer to the last question is more than a little critical28. Inequality As I indicated previously, early on elementary students are taught that one natural number is greater than another if it is further to the right on the number line. This type of comparison is impractical for large natural numbers and, in order to determine whether one of two such natural number is greater than the other, students are taught to compare digits of the two natural numbers beginning at the left. For instance if one has two natural numbers a = anan-…a1a0 and b = bnbn-1….b1b0 then a >b if, counting from the left, the first ak such that ak≠bk is such that ak > bk. For example 1245 > 1238 since 4 in the tens-place is greater than 3 in 28
And is very much related to problem three above regarding the 112 sticks.
25 the tens-place (1 in the thousands-place and 2 in the hundreds-place being equal in the two numerals). A proof of that this procedure works is rather straightforward29, but can be facilitated by an extension of the proof I derived earlier (that is, Lemma I): Lemma II Let a, b, c, and d be natural numbers such that a 10b1 + b0 Hence by transitivity 10a1 > 10b1 + b0 and as a0 ≥ 0 by Lemma II 10a1 + a0 > 10b1 + b0
27
So assume the comparison procedure works for all n ≤ N and consider the procedure when n = N + 1. Consider n = N + 1 Case I ( N ≥ k ≥ 0) Consider the two numerals aNaN-1…a1a0 and bNbN-1….b1b0. By the induction hypothesis aNaN-1…a1a0 > bNbN-1….b1b0 Further by hypothesis 10N+1aN+1 = 10N+1bN+1 so by Lemma II aN+1aMaN-1…a1a0 > bN+1bNbN-1….b1b0 Case II (k = N + 1) By hypothesis aN+1 > bN+1. Hence aN+1 ≥ bN+1 + 1 and 10N+1aN+1 ≥ 10N+!bN+! + 10N+1 Now 10N+1 – 1 is the N-digit numeral cNcN-1…c1c0 where ci = 9 for 0 ≤ i ≤ N). Compare cNcN-1…c1c0 and bNbN-1….b1b0. There are two possibilities: bi = 9 for 0 ≤ i ≤ N. that is, cNcN-1…c1c0 = bNbN-1….b1b0
28
or bi ≠ 9 for some i (0 ≤ i ≤ N) that is, by the induction hypothesis),
cNcN-1…c1c0 > bNbN-1….b1b0 Hence, in general, cNcN-1…c1c0 ≥ bNbN-1….b1b0 and, by transitivity, 10N+1 > bNbN-1….b1b0 By Lemma II 10N+!bN+! + 10N+1 > bN+1bNbN-1….b1b0 Thus by transitivity and by Lemma II aN+1aMaN-1…a1a0 > bN+1bNbN-1….b1b0 Large Numbers This section needs to be largely rewritten and the discussion of Moser moved to, perhaps, the chapter on infinite numbers. The rewrite might go as follows:
29 Out of counting grow natural questions about big numbers. How big is a billion? What is the largest number you can write? What is the largest number you can know? Do numbers go on forever? Through the use of zero and commas children learn how to write and say a billion30 1, 000, 000, 000 However, few experience such words as one of the counting numbers. Typically, children experience large numbers as powers of 10.; for instance 10 tens make 100 and 10 millions make 1 billion. That is, one speaks of magnitudes of objects in terms of powers of 10; for instance, 102 represents 100 and 109 represents 1, 000, 000, 000. Base-ten blocks are a set of manipulatives that can help children visualize with some imagination31 relatively large magnitudes. They come in four sizes: a single cube often called a one or a small cube; a bar composed of ten small cubes often called a stick or a ten; a square composed of 10 sticks often called a flat or a hundred; a single cube composed of ten flats often called a large cube or a thousand. I have seen teachers take the large cubes and begin constructing on the classroom floor the outline of a cube that that would consist of one thousand large cubes; that is a million cube. I, during discussion of these materials, often hold up the small cube and ask my students—teacher candidates—to imagine that this is a thousand cube. I then hold up the large cube and ask tor the magnitude of the large cube. This approach also allows one to imagine small magnitudes such as 10-3 ; that is, one holds up the large cube, tells students to imagine that this is a one cube, and then holds up the small cube and asks for the magnitude of the small cube. There is a quite good video titled Powers of Ten that provides similar sorts of visualizations and a number of books written principally for the elementary and middle school grades. I can recommend One Grain of Rice32 by Demi and Really Big Numbers33 by Richard
30
The use of separators varies among cultures. In the Indian numbering systems the quantity 505,000 would be written as 5,05,000. In East Asian numbering systems 505,00 would be written as 50,5000. 31 A critical factor in the doing of mathematics 32 This book considers only powers of two.
30 Evan Schwartz. The latter book provides—significantly better than my original attempts in this section—a gradual approach to something moser like. Schwartz begins his book much as I have just done; that is, he encourages his readers to visualize arrays; for example dots on a page, mounds of sand, or New York City filled with basketballs to the height of a man. He quickly moves into powers of 10 and has a nice section on names of things like 10153, a quinquagintillion. This is followed by a section where he begins to relate large numbers to physical obects; foe example, the number of carbon atoms in a pencil and onward toward galaxies and galaxy clusters. The final section of the book the introduction of a symbol similar to that introduced by moser; that is
In particular this allows him to write
and nesting boxes, in moser fashion, gives the possibility of writing really big numbers. In any case, what is really interesting about powers of ten is that it permits us to write base-10 numbers in a very compact fashion. For example ‘six hundred, thirty-four can be written 6 x 100 + 3 x 10 + 4 or 6 x 102 + 3 x 10 + 4 or
33
This book slowly works its way up toward the moser I previously took up in this section; however, it is far more gradual and far more interesting.
31 6 x 102 + 3 x 101 + 4 x 100 This last expansion,34 as I have previously indicated, was crucial in my students—teacher candidates—study and analysis of algorithms for arithmetic operations within base-10.
34
I note that very similar representations work in other bases.
32
Sums: Comments My focus in this chapter is on the mathematics of addition. This is another chapter that found its way, in part, into my teaching of mathematics methods more generally. I originally, when discussing addition, tended to jump to the complexities of multi-digit addition as this is where many of my students—teacher candidates—had not sufficiently thought through their teaching of such. However, over time, I found that many also had not sufficiently thought through the teaching of basic addition facts; especially those termed ‘making ten facts.’ Addition from a Developmental Perspective In U.S. Schools, students, as I have mentioned, begin at the end of first grade and on into second grade committing to remembrance addition facts within 5, 10 and 20. Each of these stages is of importance; however, the so called ‘making ten facts’ turn out to be critical. Consider 5+7 I know that 5 + 7 is more than 10. How much more? Well, 5 + 5 = 10 and I have 2 left over. Hence, given that we are working in a base-10 system, it must be the case that 5 + 7 = 12 I am not arguing here, as I tell my mathematics methods students over and over again, for some trick in place of memorization. However, I am arguing that memorization (I prefer the term remembrance) comes with the experience of recurring success and I am arguing that 10 in a base-10 system plays a critical mathematical role in those recurring successes. That is, although I may initially compute 5 +7 more comfortably as 5 + 5 + 2—and, hence, 10 + 2— the time will come when I realize that the sum is just 12, a ten with 2 ones. Such realization is unlikely, and I speak for some children I have taught, if 5 +7 forever remains some sort of memorized mystery.
33 Whole Number Addition Algorithms Consider this instance of the standard addition algorithm
This is essentially a streamline version of the following sequential operations: 153 + 3 156 156 +70 226 226 +200 426 Notationally, the economy of this algorithm can often cause young children some problems. However, if a child is aware that 273 = 200 + 70 + 3 The algorithm, itself, makes a lot of sense. Unfortunately, the algorithm can be introduced without much sense making. It is a mathematical entity that deserves to be studied by both student and teacher. As a case in point , I rewrite the sum of 153 and 273 as follows
Thousands
Hundreds
Tens
Ones
34
1
1 2
5 7
3 3
4
2
6
That is, I add the one, hundreds, and thousands respectively35. Hence proper use of the algorithm requires that the addends and the sum be precisely lined up in place value order. This, I emphasize, is no minor matter for students whose coordination is yet developing. Also note that doing additions on such a grid makes it reasonably clear that one is not adding, for example, 5 and 7, but 50 and 70. Other Bases Over the years I have noticed that the teacher candidates I work with have much the same difficulties with addition in bases other than base-ten that young children have with base-ten addition. In consequence, I have begun to wonder36 whether a large portion of the difficulty young children have with the arithmetic operations isn’t simply due to not having sufficient time to develop a substantial degree of fluency in our base-ten number system. So as to illustrate for my mathematics methods students the difficulties their students will face—difficulties they themself once faced, but have now forgotten—I often pose computational problems within the base-6 number system although it is possible to pose similar problems in all bases (base-2 is somewhat of a special case and base-16 causes useful consternation). Basic Addition Facts
35
Students should able to mentally compute simple sums such as 345 + 300 or 245 + 20. A sure sign in the upper elementary grades that a student has significant problems with multi-digit addition is the student using the standard addition algorithm to compute such sums. 36 I think of the teaching experiment attempted by Louis P. Bénézet in this regard.
35 All of the teacher candidates I work with have no problems with their addition facts within 6; however, a sort of panic ensues when I first37 ask them to do a sum similar to the following38 56 + 36 Some39 pull out base-6 charts they had made during our discussions of counting; some pull out a base-6 number lines; some even convert to base-1040,compute, and then convert back to base 6; however, relatively few remark, “Well, you take one from the three to make six out of the five and you have two left over, so the sum is 126.” It is, as if for them,10 has more verbal significance—it acts as a sort of flag—than mathematical significance. Multi-digit Addition I often tell the teacher candidates in those mathematical methods courses I teach that, for instance, the algorithm for base-6 multi-digit addition is essentially the same as that for multidigit base-10 addition; the only real difference being that the columns have place-values of powers of 6. A few take me at my word; many of the others continue to treat such addition, much like many of their own students will do, as a mystery. What I find fascinating about all this is that mistakes made by these teacher candidates closely resembles those made by elementary school students in their first attempts with multi-digit base-10 addition. Some teacher candidates, for instance, appear to be unsure how to handle the carry and, at times, forget to carry. While many of these teacher candidates often mention to me that they are now, due to such exercises, most take away an awareness of the mathematical difficulties elementary school 37
I consider such sums after we have had the lesson on converting between base-10 and base-6 and I have given a sort quiz on such conversions. 38 The subscript ‘6’ indicates these numbers are in base-6 39 I happen to be fine with students initially using number line base charts and even converting to and from base-ten (which I consider equivalent to counting on their fingers). 40 This act, I tell the teacher candidates, is on a par with a young child putting their hands behind their back and counting on their fingers. An act that needs to be treated with respect as it is, I think, rooted in a degree of discomfort with what a child may view as some untested method of addition. Such children should be encouraged to take on numbers that do not lend themselves to finger counting.
36 children may face. Thinking back to these moments, I now regret not having asked teacher candidates to describe their thinking when doing such addition. Such descriptions might have been a foundation for deeper discussions on how to effectively teach multi-digit addition. Teaching mathematics, I would suggest, is fundamentally hermeneutic and involves a toand-fro between practice and theory. Former colleagues at the University of Michigan appear to be usefully involved in immersing teacher candidates in the ongoing efforts of elementary school children to perform, for instance, multi-digit addition. However, my sense is that they are less involved in helping teacher candidates expand their mathematical awareness of the entailments of such performances. Packs and Pallets The packing of bottled water as introduced in the previous chapter offers a somewhat concrete way of making sense of the standard addition algorithm in base-641. For example, assume that I am preparing a shipment of 4 pallets, 2 tiers, 3 six-packs, and 2 bottles for Cub Foods and receive a call asking if I can send an addition 5 tiers and 5 bottles. What do I do? Well, 2 bottles plus 5 bottles is 7 bottles and that is 1 pack and a bottle. So now I need, at least, 4 pallets, 2 tiers, 4 six-packs, and a bottle. However, I have 5 additional tiers to add and this gives me a pallet and a tiers. So the final order consists of 5 pallets, 1 tier, 4 sixpacks, and a bottle. That is
Notice how the standard addition algorithm in base-six captures such reflections.
41
I note that something similar is often done with bean sticks or base-10 blocks in the classroom
37 Addition Estimates Using the Common Core Standards as indicative, I see little mention of estimation until third grade. In so far as addition is concerned, a lesson in third grade often takes the following form: The teacher puts an addition problem on the board—for instance 154 +879 and asks students to estimate the sum to, for instance, the nearest hundred. Both teacher and students often seem to be confused about what pragmatically42 this means. Does one, for instance, estimate the addends to the nearest hundred and then add or—and this unfortunately become the default—does one add as usual and then estimate to the sum to nearest hundred43? This, I think, is an unfortunate state of affairs. Estimation, one might argue, is one of the number senses and, pragmatically speaking, it is a mathematical virtue that can be invaluable in dealing with the real world. Partly due to such concerns, partly because I wish to revisit base-ten, and partly because I wish to make the point that school algorithms are not necessarily the most efficient, I sometimes introduce the scratch addition algorithm. The scratch addition algorithm is ideally suited to estimation as addition begin on the left.44 For instance, adding from the left gives 154 +879 42
As, in a sense, am I If this is the interpretation and one listens closely, one often hears from a student, “Why are we doing this? I don’t need to estimate, I have the answer.” 44 If you think about this for a moment, you may realize it is very difficult to estimate the magnitude of a sum if you base your estimates on additions from the right. 43
38 923 103 Steps are as follows Step I: Add the hundred column getting 900 154 +879 9 [So a first estimate of the sum is about 900] Step II. Add the ten column getting 120. Add the 100 of the 120 to the 900, getting 1,000. Scratch out the 9, writing 1,000 in the appropriate column and write the 20 from the 120 in the appropriate column 154 +879 92 10 [So a second estimate of the sum is about 1,020] Step III. Add the one column getting 13. Add the 10 of the 13 to the 20, getting 30. Scratch out the 2, writing 30 in the appropriate column and write the 3 from the 13 in the appropriate column
154 +879
39 923 103 [So the sum is 1,033] Arithmetic Series and Figurate Numbers This section in the book is far too short given the wealth of material available; I greatly encourage some reading of the beautiful book by John Conway and Richard Guy45. Nevertheless as I can resist, here are some classics 1 1+3 1+ 3 + 5 1+3+5+7 ...........
= 12 = 22 = 32 = 42
1 7+9 25 + 27 + 29 61 + 63 + 65 + 67 ..............
1 3+5 7 + 9 + 11 13 + 15 + 17 + 19 ............... =14 = 24 = 34 = 44
= 13 = 23 = 33 = 43
1 15 + 17 79 + 81 + 83 253 + 255 + 257 + 259 ..................
=15 = 25 = 35 = 45
I leave it to you to add some lines or even a schema for powers of 6. If you need some hints, I recommend Extensions to a Theorem of Nicomachus in Introduction to Arithmetic by Martin Luther D'Ooge, Frank Egleston Robbins, and Louis Charles Karpinski. Macmillian, New York (1926).
45
John H. Conway and Richard K. Guy. The Book of Numbers. Springer-Verlag, New York (1996).
40 Indeterminate Problems
I admit to having a fondness for intermediate problems and that, I am sure, influenced their inclusion. My goal, nonetheless, was to raise the awareness that so-called guess and check problems having multiple solutions can often be handled in, what I view, an aesthetical mathematical fashion. That is, I wished to give these teacher candidates some awareness of what, I see, as the sheer power and beauty of the mathematical method. Augmented Problem Set 1. If order is irrelevant, how many different addition facts would a child need to memorize in order to (a) make 105; (b) make 106 2. After holiday shopping, a young NY resident finds that she cannot pay her January rent. She proposes to the landlord that, on each of the (31) days of January that she has not paid her rent, she will give him one of the links of her 18 karat gold necklace, which just happens to have 31 links. Then, at the end of the month, when she provides the delinquent rent, he will return all of the necklace pieces. After duly verifying the quality of the necklace, the landlord agrees. But then the woman becomes concerned over the cost of reassembling the necklace at the end of the month. After some thought, she has an idea. On the first day, she will cut off one link to give him. But then, on the second day, instead of cutting off another link she will cut off a pair, give him this 2-link piece, and retrieve the one link that he has. Then, on day three, she will return the single link, so that he then has the required three links. And so on. Now the problem is: What is the minimum number of cuts necessary so that on each of the 31 days of January, she can give the landlord the exact number of links required on that day?
41 An addition table in base-10 would look something like this + 1 2 3 4 5 6
1 2 3 4 5 6 7
2 3 4
3 4 5
4 5 6
5 6 7
6 7 8
7 8 9
8 9 10
9 10 11
And so on ….
3. Derick wants to learn his addition facts in base-5. Give him a hand and create a base-5 addition table. 4. Sajeda wants to learn her addition facts in base-2. Give her a hand and create a base-2 addition table. 5. (optional) Giselle wants to learn her addition facts in base-16. Give her a hand and create a base-16 addition table. 6. Compute 2347 + 6567. Make sure to justify your addition of single digits. 7. Let T=10 and E =11. Compute TT1912 + EE2212. Make sure to justify your addition of single digits. 8. Compute using the scratch method (a) 2356 + 456; (b) 4479 + 4672 9. Use a variation of Gauss’ approach to sum the (a) even numbers to 100; (b) the multiples of 3 to 100. 10. Prove that the last term in a series of ‘n’ consecutive odd integers beginning with 1 is 1 + 2(n-1) 11. Prove that a series of ‘n’ consecutive odd integers beginning with 1 sums to n2 (Hint: use #9 above and the expression I developed for the sum of consecutive natural numbers in this section of the book.
42
Differences: Comments Much of my focus in this chapter was on the mathematics done by children in the primary grades and early upper elementary grades. I begin with a brief sketch of ancient subtraction practices and then move to a discussion of standard subtraction algorithms. This is followed by an attempt to make sense out of what, I consider, the confusion teachers and students bring to negative numbers. I note that subtraction causes students a number of difficulties46: ⁃
It is, in an essential sense, not the opposite of addition. Consider when I count-on on my fingers or the number line, I begin with the finger that is right-adjacent to that just counted and end on my answer47. That is, 5 + 4 is 1 2 34 56789 12345 1234 However, when I subtract, I begin with the last finger counted and my answer is one digit
to the left of where I end48. That is 5 - 3 is 1 2 3456789 123 ⁃
It gives rise, in a sense49, to the extension50 of numbers termed integers which, while including the natural numbers, adds additional structure and, at times, gives rise to
46
Some students never regain their footing due to these difficulties This somewhat gives rise to the notion of addition as incrementing 48 This somewhat gives rise to the notion of subtraction as canceling 49 Consider the solution to an equation of the form 47
x +10 = 5
43 nonsense; for example, ‘Jon has 5 marbles, if he gives 10 to Judy how many does he have left?’ Subtraction from a Developmental Perspective There seems to be a curricular rush to finish up addition and take on subtraction. The vignette I include in the book illustrates, to a degree, how children look at this rush. At this point in the curriculum they have, for the most part, become quite adept at counting on and, hence, it should be no surprise they solve what adults might see as a subtraction problem ?+4=5 by counting on; that is, they operationalize subtraction51 as difference. This is no minor manner and something, I think, that should be encouraged as the mathematical notion of difference seems more robust that that of counting-backwards. Whole Number Subtraction Algorithms Base-10 Decomposition Subtraction Algorithm This is what we, in the US, term the standard subtraction algorithm. Consider, for example, 1245 -789
All coefficients and constants are natural numbers; however, the solution is not. 50
Most teachers do not, unfortunately, indicate that, in a sense, the understood rules of classroom mathematics have changed. 51 I am not talking about counting-backwards
44 This can be step-wise computed as 1245 -
9
1236 1236 - 80 1156 1156 -700 456 Or, more elegantly, as
I note, for example, one does NOT subtract 9 from 5, but subtracts 9 from 1245. Further, the notation used in the algorithm has, it seems, given rise to the myth of borrowing. Rather it is the case that on, for example, decomposes the four tens into 3 tens and ten ones. I have noted previously teachers and students need to study such algorithm as this is a polished piece of mathematics. Again, one way to get an appreciation of this algorithm is to rewrite in—as I did in the chapter on sums—on a place value grid. Besides making the algorithm much more transparent this will reinforce the continuing importance of place value and step the stage for the part it will play in the algorithms for multiplication and division. Equal-Additions Subtraction Algorithm
45 In this algorithm the concept of difference becomes central; that is, 1245 - 789 = 1245 + x - (789 + x) or, in particular52, 1245 - 789 = 1245 + 10 - (789 + 10) = 124(15) - 799 Algorithmically
Left to Right Subtraction Algorithm This algorithm, much like that I discussed under Sums, provides a running estimate. For example,
Base-6
52
Note I an using an addition ‘trick’; that is (10 + x) - x = 10
This ‘trick’ is well worth remembering as it is used time and time again in matters of mathematical thinking.
46 The standard subtraction algorithm in any base is structurally much like that in base10. Again, my inclusion of a discussion of the subtraction within base-653 was primarily to encourage students within my mathematics education classes to carefully re-examine the standard subtraction algorithm. As with the standard addition algorithm these teacher candidates have difficulties that were similar to those students display in elementary school mathematics classrooms. One critical question I neglected to ask at this juncture was the following: Comment on a difficulty you had with subtraction in base-6 that give you better insight into what children, dealing with subtraction in base-10, experience in the elementary school classroom. Packs and Pallets As with the standard addition algorithm, the language of pallets and packs provides a somewhat more concrete way of studying54 the standard subtraction algorithm in base-6. Consider
My story line might go as follows: Hill City has ordered a tier, 2 six-packs, and 3 bottles. However, I get a call just prior to sending the order out, that the principal just realized that is 4 six-packs and 5 bottles too much. Assume that I only have one tier, 2 six-packs, and 3 bottles on hand. What do I do? Well, I break open of the six-packs which gives me 9 bottles and I put 4 of those bottles on the truck. Then I break open one of the tiers which gives me 7 six-packs (remember I just broke open a sixpack) and I put 3 of those six-packs on the truck.
53
And other bases Base-10 blocks and bean sticks provide similar affordances for students in the elementary grades. 54
47
Augmented Problem Set 1.
If order is irrelevant, how many different two digit subtraction facts within 20
would a child need to memorize in order to (a) make 5; (b) make 6 2.
Three men go to stay at a motel, and the man at the desk charges them 30.00 for a
room. They split the cost ten dollars each. Later the manager tells the deskman that he overcharged the men, that the actual cost should have been $25.00. The manager gives the bellboy $5.00 and tells him to give it to the men. The bellboy, however, decides to cheat the men and pockets $2.00, giving each of the men only one dollar. Now each man has paid $9.00 to stay in the room and 3 x $9.00 = $27.00. The bellboy has pocketed $2.00. $27.00 + $2.00 = $29.00. So where is the missing $1.00? 3.
I claim that you can subtract any three-digit number from 1000, by subtracting the
hundreds and tens digit from 9 and the units digit from 10. Prove that I am correct (You need to argue logically rather than just give examples). 4.
Subtract (a) 7548 – 3228; (b) 7328 – 3578
5.
Mr. Wall buys a goat for $25. After she eats the dahlias, he sells her for $35.
When hearing of this, Mr. Carlson, a neighbor, tells him that he really liked that goat and, if he had known Mr. Wall was selling, he would have paid $55. Mr. Wall buys the goat back for $45 and sells her to Mr. Carlson for $50. Did Mr. Wall break even, loose or make money. If he lost or made money, how much? I note that #2 is from the book and, although teachers enjoy the problem (as do their students) it has minimally to do with subtraction. #3 seems a little difficult for many and
48 teachers usually argue three different answers for #5 (I usually let this problem hang for a week).
49
Multiples: Comments My focus in this chapter is the multiplicative structure of the natural numbers. I find it rather fascinating that a simple shorthand notation for repeated addition, for example 5+5+5=3x5 has profound mathematical consequences. In fact in writing this book, I spent considerable effort in searching out problems that were purely additive; efforts that were largely unsuccessful. Interesting problems—and this may reflect a bias—largely entailed a degree of the multiplicative. Weaving together the additive properties of number together with the multiplicative properties of number produces a rich tapestry that continues to inspire mathematics research. A recent conjecture—which is accessible to teachers and elementary school students alike—is the following55: Take any natural number. If it is 1. even, divide by 2 until it is odd or until 1 is the result 2. odd, multiply by 3 and add 1 and begin the process all over again It is conjectured that eventually the result will be 1. Multiplication from a Developmental Perspective Children, I note, most usually come to elementary school classroom with notions of counting and reasonably soon they become proficient within the naturals in the beginnings of
55
The Collatz conjecture.
50 addition. Hence, it is not by chance that multiplication is defined as sort of repeated addition— for example,
3x4=4+4+4 taking the form of repeated addends or a sort of skip counting56. However, early on, teachers also hear from their students’ language such as “double” and “triple”; a language that has the potential of evoking a sort of scaling of the number line. These notions of multiplication, I note, culminate—formally, that is—in the notion of a binary operation called the product; an entity that is visually tabulated by means of what is termed the multiplication table. Repeated Addition Within the natural numbers multiplication can be defined57 thusly 1xa=a
(D1)
(n + 1) x a = n x a + a where ‘a’ and ‘n’ are natural numbers. From this definition it follows that for any natural numbers ‘n’ and ‘a’
n
nxa =
∑a
(D2)
i=1
that is, repeated addition. In fact, it follows that for ‘n’, ‘a’, ‘b’ naturals that n x (a +b) = n x a + n x b
56
For instance, something like “two, four, six, eight ...” See, for example, Leon W. Cohen and Gertrude Ehrlich, The Structure of the Real Number System, Princeton, D. Van Nostrand, p.23. 57
51 That is, the distrubutive law holds over the naturals.
Augmented Problem Set 1. Prove (use induction on ‘n’) that definition (D1) implies
n
nxa =
∑a i=1
for ‘n’ and ‘a’ natural numbers. 2. Prove (use induction on ‘n’) that definition (D1) implies n x (a +b) = n x a + n x b for ‘n’, ‘a’, ‘b’ natural numbers.;that is the distributive law.
Scaling It may be possible, as I have indicated, to build an understanding of multiplication that owes much to the notions of doubling and tripling that students bring to the elementary school classroom. Barry Mazur58 suggests that one view multiplication by a positive integer within the naturals as effectively a scaling of the positive number line59. That is 2x
58
Barry Mazur. Imagining Numbers: Particularly the Square Root of Minus Fifteen. Farrar, Straus and Giroux. New York (2003). 59 See also the unpublished manuscript of Raymond Hoobler. A Mathematician Looks at Numbers. August 20, 2004.
52 is, in effect, an function60, mapping the naturals onto the naturals; for example, 3 to 661 [2x](3) = 6 I note in apassing that Mazur’s geometrization of multiplication provides useful visualizations of multiplication by integers, fractions and imaginary numbers. Demonstrating associativity, commutativity, and the distributive rule is a little awkward on paper62; however, the use of materials such as string or Cuisenaire rods makes things a bit more straightforward. Imagine, for example, I have taken some 1-unit Cuisenaire rods and stretched them into 2-unit rods ! I then pile up three of these 2-unit rods63
and compare that array to one built from two 3-unit rods stretched from 1-unit rods as above from 1-unit rods
60
A single column of the multiplication table provides a nice tabular eillustration of such patterning that visually accords with the functional notation I introduced previously in the chapter on Counting and Recording of Numbers. 61 I have used brackets [] to, in effect, isolate the function rule. 62 A possible way of approaching associativity is to argue, for example, that stretching the number line by 2 and then by 3 is equivalent to stretching it by 6. 63 I could concatenate these rods and demonstrate equality; however, I have piled them so as to make connections with some later comments of mine. As to preferences, I note that arrays are a little easier to draw and require significantly less precision. However, the Cuisenaire rods (or appropriate lengths of string) have definite advantages.
53 This, as I note later in this commentary, is a reasonably convincing demonstration that [2x](3) = [3x](2) What is most interesting64 is what shows up naturally when I campare the original number line with ones formed from [2x] and [3x]. Consider
0
1
2
3
4
5
6
7
[2x] 0
1
2
3
4
[3x] 0
1
2
3
Note, thus far, 1. the only place the stretched number llines simultaneously ine up with a natural number on the original number line is 6. 2. The [2x] number line only lines up with with a natural number on the original number line at 2, 4, 8, 10 3. The [3x] number line only lines up with with a natural number on the original number line at 3, 6, and 9 That is, Mazur’s scaling seems to naturally bring multiples and factors to light.
64
I note that considerations of scaling in one-dimension leads nicely to a consideration of scaling in multiple dimesniosn and, hence, to scaling, for instance, of area and volume.
54
Augmented Problem Set 1. Demonstrate by concatiinating—that is, building trains—Cuisenaire rods (or a rope using string) the associative rule 2. Demonstrate by concatiinating—that is, building trains— Cuisenaire rods (or rope using string) the distributive rule. Multiplication Facts Base-Ten Facility with multiplication is developed in ways that somewhat resemble those used for developing facility in addition. That is, students are expected to remember multiplication within 10 and then use such remembrance in performing multi-digit multiplication. This, in a sense, is no minor task as, on the surface, it seems to require committing to remembrance 100 facts; that is, the multiplication table. However—and I stress this however—these facts are not unrelated and reflection in this regard on the part of a teacher can considerably simplify the task. I usually begin discussion on these relations by pointing out that among the first things a child needs to remember are three rules65: Rule
Example
Commutative Rule
2x3=3x2
Associative Rule66
2 x 3 x 4 = 2 x (3 x4)
65
I stress how essential it is that children understand (within reason) and have facility with these rules. 66 The associative rule has an important corollary; that is a x b = (a x c) x (b ÷ c) For instance,
55 = (2 x3) x 4 Distributive Rule
2 x (3 + 4) = 2 x 3 + 2 x 4
The first and third rule can be demonstrated fairly convincingly with a two-dimensional array67. For instance, 2 x 3 can be represented by
and 3 x 2 by
The second rule—as it involves three terms—can be demonstrated with a three-dimensional array68. I then point out that the commutative rule reduces the number of multiplication facts to be remembered from 100 to 55. Further both 0-times facts and 1-times facts are easily learned, and this reduces the number of multiplication facts that might pose challenges to remembrance to 36. Of these the 2-times facts normally pose little difficulty—especially if children have been
7 x 12 = (7 x 6) x (12 ÷ 6) = 42 x 2 67
That is, 3 x 5 who be represented by an array with three rows and 5 columns I have seen, for instance, teachers posing problems that require, in effect, children to fill a box; for example, the operation 3 x 4 x 5 would be performed within a box of dimensions 3” by 4” by 5” with square inch cubes. 68
56 exposed to skip counting— and this reduces the number of multiplication facts that might pose challenges to remembrance to 28. The 4-times multiplication facts only require a doubling of the 2-times multiplication facts; for example69 4x8=2x2x8 = 2 x (2 x 8) and the 8-times facts are almost as easy. For example 8x7=4x2x7 = 4 x (2 x 7) = 2 x 2 x (2 x 7) = 2 x (2 x (2 x7)) Mastery of these multiplication facts reduces the number of facts that might pose challenges to remembrance to 15. The 3-times multiplication facts can be handled with the distributive rule; that is, for example 3 x 6 = (2 + 1) x 6 =2 x 6+6 and 6-times facts are just the doubling of 3-times facts. That is, for example70, 6x7=2x3x7 = 2 x (3 x 7) 69
My point is not that the relations I note here are the way to teach the multiplication table, but that there are ways to remembrance that do not initially require blind memorization. It is critical to build on what children know, not what they don’t know. 70 I remember as a child always having trouble with 7 x 6 and, for many years, I remembered it by using the distributive rule; that is I remember that 6 x 6 = 36 and thus 7 x7 was just 36 + 6 = 42.
57
Mastery of these facts reduces the number of multiplication facts that might pose challenges to remembrance to 7. Children, most usually, know how to skip count by 5 and many know some trick for 9-times multiplications fact71s. Mastery of these multiplication facts reduce the number of multiplication facts that might pose challenges to remembrance to 1; that is 7 x 7. This last multiplication fact is, of course, just one of the squares. I emphasize to my students—in-service and pre-service elementary school teachers ▪
that it is possible to make such remembrance doable by segmenting what their students need to master.
▪
that remembrance should be done within some context that makes it of value. I for example, do not remember my home address because I have said it over and over to myself. I remember it because I am occasionally asked for it
▪
that somehow children need to reasonably rapidly access such facts from their memories either as a simple multiplicative product or some mathematical identity—for example, I know 6 x 7 = 42 because I know 6 x 6 = 36 and 6 x 7 is just one more 6.
I note that, most usually, children, who initially learn such products through some sort of mathematical identity, go on later to internalize the product. I also note that children who have difficulty in mastering the multiplication table, most usually, struggle with mathematics from this point on. Other Bases I do ask my students to construct multiplication tables in other bases. However, my purpose in this is not to engage them in some sort of memorization, but to have them invest some time in studying the structure of a multiplication table; a structure that can, reasonably easily, be generated through skip counting and the commutative rule.
71
For instance, the product of 9 x n (n a digit greater than 1) is the number with tens digit n-1 and ones digit 10-n). Thus 9 x 7 is just 63. I sometimes ask my mathematics education students to prove that this shortcut works. This result is a nice application of the distributive rule.
58 Multiplication by Powers of 10 Understanding, at least in a pragmatic fashion, multiplication by powers of ten72 is essential to making sense of multi-digit multiplication and division. Unfortunately, most students in my methods classes have learned the language of adding on zeros (or, in effect, removing them when dividing by a power of ten) and, even more unfortunately, communicate with their students in this fashion. There is a much more robust73 way of looking at this operation. Consider 10 x 142 as 142 = 1 x100 + 4 x 10 + 2 x 1 The distributive rule gives 10 x 142 = 10 x (1 x 100 + 4 x 10 + 2 x 1) = 1 x 1000 + 4 x 100 + 2 x 10 = 1420 That is, multiplication by 10 just shifts 142 one place to the left—that is, toward increasing74 magnitudes—on the place value scale75. Similar considerations hold for multiplication by 100, 1000, and so on76 as a base-ten number system, by definition, naturally supports such shifts. 72
Students should develop facility in mentally multiplying (or dividing) by simple powers of ten—for instance, ten, one hundred, and one-thousand. Students (or teachers) who utilize the standard multiplication in performing such computations will find themselves seriously struggling with much of the school mathematics curriculum from this point on. 73 And a way that naturally extends to decimal multiplication. 74 Some of my students remark, at this point, that the decimal point moves. While this functionally is an effective mnemonic, (1) there is no decimal point in this problem; and (2) the decimal point moves towards decreasing magnitudes. 75 That is,
59 So as to emphasize that the language of adding zeros is not particularly helpful (and, sometimes, as a lead into a discussion of decimals) I often pose a problem much like the following for my mathematics methods students
Take any number 1.
Add a right hand zero
2.
Add one hundred
3.
Divide by five
4.
Subtract ten
5.
Subtract ten
6.
Half
I conjecture you will have the number you started with. •
Check some numbers to see if I am right or wrong.
•
If I seem to be right, show why this always works and write a different problem of your own that always works.
•
If I seem to be wrong77, fix my conjecture and show why the ‘fixed’ conjecture always works. Write a different problem of your own that always works.
……. 1000 100 10 1 .1 .01 .001 .. Clearly multiplication by 100 results in a shift of two to the left—that is, toward larger place values— while multiplication by 1000 results in a shift of three to the left 77 In case you think otherwise, I am wrong. 76
60 Whole-Number Multiplication Algorithms Base-Ten If a student is able to access their multiplication facts reasonable quickly, understands multiplication by powers of ten, and understands the distributive rule, the standard multiplication algorithm makes a reasonable amount of sense. Additionally the standard multiplication algorithm employs the notion of shift I introduced above. For instance 28 x 142 would appear as
That is, since the second row of the algorithm involves multiplication by 20, I shift all products one place value—that is, I multiply 10—as I multiply by 2. Further note that place value alignment is crucial in performing the algorithm correctly. Students that have difficulties in maintaining alignment should be encouraged do such problems on gridded paper. There are, I note, intermediate multiplication algorithms that aid in such sense making; for instance, the lattice method and the array method. As I consider the lattice method in the book (and, in a different guise, the array method), I will only discuss the array method herein. The array method is just an extension of that array representation I introduced earlier for visualizing single digit multiplications. So 28 x 142 can be rewritten as
100
40
2
20
2000
800
40
8
800
3200
16
61 Note that the sum of the second row in the array is just the first row in the standard multiplication algorithm, while the sum of the first row of the array is just the second row in the standard multiplication algorithm Other Bases While I do ask my mathematics methods students to perform multiplication in other bases, my focus78 is more on whether they appreciate the role the base plays rather than becoming facile79 in single digit base multiplications. For instance, in base-six, I would expect the following reasoning while using the standard multiplication algorithm for 1226 x 246
That is, proceeding in the usual fashion from the right to the left, I multiply the 4 in ones place in 246 times the 2 in the ones place in 1226 which, in base-10, gives me 8. However in base-6 that is 126 so I write down the 2 and carry the 1. Then I multiply the 4 in ones place in 246 times the 2 in the six place in 1226 which, in base-10, gives me 8 and I add the 1 I carried previously for 9 in base-ten. However in base-6 that is 136 so I write down the 3 and carry the 1.Then I multiply the 4 in ones place in 246 times the 1 in the thirty-six place—that is the 66 place—in 1226 which, in base-10, gives me 4 in base-ten and add the 1 I carried previously for 5 in base-ten. However, this is just 56. I now perform a series of similar operations for the 2 in 246 keeping in mind that I must shift the second line one place value to the left.
78 79
This also sets up my discussion of congruences in the next chapter of the book. This might be worthwhile, but given time constraints was never practical.
62 Prime Numbers and Factoring Noticing the properties of numbers begins in the early grades with discussion of zero and even and odd numbers and continues into the upper elementary grade with discussion of prime and composite numbers. Students, if given the opportunity, have already begun to notice that individual numbers are interesting. This noticing will serve them well throughout the grades. Prime Numbers Prime numbers still fascinate both children and mathematicians and, in a sense, are the building blocks of the multiplicative structure of mathematics. There is no easy way to determine whether a number is prime although it is relatively easy to prove there are an infinite number. However, I discuss using Eratosthenes’ sieve to check whether a number N is prime as one need only consider factors up to
.
One problem I give to get my mathematics education students to get them thinking about primes is Susie comes up to your desk. "Mr. Bass," she says, "I have a conjecture. We've been talking about primes. Y'know those numbers which are only divisible by themselves and one. Like nineteen! I've been experimenting. I don't think any of the numbers between one and eleven are differences of two primes." Mr. Bass thinks for a moment and says, "What about three minus two?" Susie frowns, "I thought you said we didn't have to do one, but anyway I said 'between one and eleven.' Also I think this is the only one." Mr. Bass smiles and asks, "Is this another conjecture?" Susie smiles and replies, "Not yet. I need to experiment more." Mr. Bass says, "Okay. How about five minus three." Susie frowns and says, "I forgot about two because it is so weird. An even prime! But I'm pretty sure about the rest [she sounds unsure]. Oh no! I just realized that twenty-three minus thirteen is ten! I guess it isn't a good conjecture." Mr. Bass smiles. "Perhaps we should get the rest of the class to help. Okay, how about something like [and he writes and says]
63 Susie's Question What numbers between 1 and 11 are not the difference of two primes? Prove your answers. Then he smiles and says, "Do you want to include that business about one." Susie giggles, "Yes!" Mr. Bass adds
Susie's Conjecture The only time 1 is the difference of two primes is when the primes are 2 and 3 (2 minus 1 doesn't count). Prove your answer. Give Susie and her class a hand and make sure you prove your answers. Prime Factorizations One activity that gets children thinking about prime and composite numbers is as follows: On the black board draw or pin the following figures80:
Then ask students to assign values and ask them to describe what they notice81. As seat work ask them to draw similar diagrams for 10 through, at least, 12. 80
This assumes a prior understanding of multiplicative arrays
64
I discuss in this section a reasonably straightforward technique for determining all the factors of a number that draws on the combinatorial notions I presented in the chapter on counting. However, I have found that my mathematics methods students are still somewhat unsure about such notions and that I need to reinforce their understanding before what I discuss here makes a lot of sense. Augmented Problem Set 1. Compute 154·34 using (a) the Egyptian method; (b) the Duplation and Mediation method; (c) the Lattice method; and (d) the YouTube method 2. What possible digits appear as the last digit of (a) a square number? (b) A cubic number? Justify your answers. 3. Create a multiplication table for base-7 4. Multiply using partial products or arrays (a) 217 by 237; (b) 457 by 237 5. Multiply using the standard algorithm (a) 3227 by 2127; (b) 4627 by 2127 6. Give a justification for YouTube multiplication or Duplation and Mediation 7. Write82 down a line of 8's. Insert some plus signs so that the resulting sum is 1000. How can this be done?
81
It is a good idea to ask, for example, whether there is another way of drawing 2 or 6 as this will highlight the commutative rule and then ask whether—disregarding commutative variations—all the diagrams have been drawn for the numbers 1-9. Interesting 8 has a three dimensional diagram. 82 This problem is due to Hyman Bass and the wording is a bit terse. Basically you are looking for all sums of 8, 88, 888, etc. that equal 1000. My mathematics education students find a few examples helpful.
65
Divisibility and Remainders: Comments
In this chapter I begin a discussion on divisibility that, in a sense, continues on into those succeeding chapters taking up fractions, decimals, and real numbers. I find it fascinating that questions of divisibility adds another rich layer to the mathematical landscape. A landscape in which children can, if given the opportunity, explore in their early mathematical years. My focus in this chapter is, to a degree, an expansion of my exploration of the multiplicative structure of the natural numbers in that one explores solutions83 to N = q•d + r for r>0 For example, it is the case that 25 = 5 x 5 However, it is also the case that 25 = 4 x 6 +1 25 = 3 x 8 + 1 25 = 2 x 12 + 1 I begin the chapter by discussing some of the early ways by which division was accomplished84. I then discuss the standard division algorithm before taking up divisibility rules. I note that I seldom in my mathematics methods courses covered the materials in this and the succeeding chapters. Hence, in a sense, what you read here has undergone less reworking than the preceding chapters.
83
Here, as traditional, q is termed the quotient, d the division, and r the remainder. I note that an argument can be made that galley method of division is more efficient than the standard division algorithm used in US schools. I also note that the US standard division algorithm is not an international standard. 84
66 Division from a Developmental Perspective In a sense the very idea of division grows naturally from mathematical questions that should arise as children become comfortable with multiples. Consider, for example, the follwoing problem and its variation Jamal has 6 cookies which he wishes to divide evenly among himself and two firiends. That is 6 cookies = 3 persons X cookies per person or Jamal has 6 cookies. He plans to take two cookis for himself and give two to each of his friends. That is 6 cookies = number of persons X 2 cookies per person The first of these variations is termed partitive division; that is, I know the original number of cookies and the number of parts into which they are to be divided. The second is termed quotitive division; that is I know the original number of cookies and the quota to be alloted each person. From such simple beginnings children move, somewhat enxorably toward the standard division algorithm. Consider 1987 ÷16
This algorithm is highly efficient and hence quite opaque to a number of students. Note
67 ▪
All digits in the algorithm are aligned according to the place values of the dividend. That is, the 1 digit in the quotient 124 is aligned above the 9 in the dividend 1987 and, hence, represents 100. The product of that 100 and the divisor 16 is 1600 and, hence, the 1 in that product is aligned with the 1 in 1987; the trailing zeros suppressed as in the multiplication algorithm.
▪
The 100, the 20, and the 4 that make up the quotient are all maximal. That is, given a division of 16, 100 is the maximal hundred that that is a quotient for 1987; 20 is the maximal ten that is a quotient for 380; and 4 is the maximal one that is a quotient for 67.
Hence, to have a modest understanding of this algorithm a student must have considerable understanding of place—so as to appreciate the necessity for careful alignment—and substantial85 facility with multiplication—so as to obtain through trail multiplications a maximal quotient. While exercising the standard division algorithm as intended will always require notational care and facility with multiplication, it is possible to give students (and teachers) considerable understanding of what is entailed in the algorithm. For instance, the manner in which US teachers write the quotient significantly contributes to its opaqueness and, in other parts of the world, teachers write, for example,
85
Students who, at this juncture in their schooling, lack facility in multiplication will be at be a substantial disadvantage when facing the standard division algorithm.
68 While this form of the standard division algorithm removes the necessity for strict place value alignment in the quotient86, some relaxed87 variation of removing copies of 15 such as the following88
removes the necessity for maximal89 partial quotients; that is the 100, the 20, and the 4 in 124. This latter version should be seen as intermediate in that it is less efficient than the usual standard division algorithm.
86
Trailing zeros may still be suppressed although the version I have illustrated retains them. I am not arguing that this relaxed version should replace the usual standard division algorithm, but pointing out that this relaxed version can aid eventual mastery of the standard division algorithm. 88 That is, division within the natural numbers can be realized as repeated subtraction just as its opposite, multiplication within the natural numbers, can be realized as repeated addition. 89 Some of the anxiety surrounding the continual computation of maximal partial quotients can be removed by an initial computation of simple multiples of the divisor. For example, the following multiples of 16 will suffice in all division problems entailing division by 16: 87
1 x 16 = 16 2 x 16 = 32 3 x 16 = 48 4 x 16 = 64 5 x 16 = 80 6 x 16 = 96 7 x 16 = 112 8 x 16 = 128 9 x 16 = 144
69 Clock and Modular Arithmetic
In part, I included this section on modular arithmetic because it seemed accessible to my mathematics education students and, in part, because notions taken up are, I think, interesting and powerful. I begin with briefly—too briefly—considering the peculiarities of clock arithmetic, giving a meaning to an expression of the form 5 ≡ 2 mod 3 that is, 5 is divisible by 3 with a remainder of 2 or equivalently (5 - 3) is divisible by 3. I then discuss the properties of this notation and give some applications among which are the characterizations of divisibility by single digits90 often taught in the upper elementary school grades and the technique of casting out nines which, with care, might be of use to suchstudents as a quick partial check on complicated arithmetic problems.
Modular Arithmetic Some years—years when I had insufficient time to discuss modular arithmetic—I have divided the class into groups and gave each group a series of sequential numbers from 1 to 50, asked them, in effect, to compute remainders of ones and tens digits and, when appropriate, asked them to repeat such computations with an appropriate sums of those remainders, and then asked them to report out on any patterns they noticed91. A completed worksheet for divisibility by 3 might look roughly as follows
90
Some of the better known are by 2, 3, and 9. This exercise can also be simplified a bit if all you want to do is focus on patterns. That is, just give each group the same series of digits and as they to add the digits in the place value expansion (and to repeaqt this as many times as necessary if the sum is greater than 9). A point I try to make here is there is a lot to see in simple explorations of the naturals. 91
70
10-
1-digit Sum
digit
10-
1-digit Sum
digit
1
1
2
2
3
0
4
1
5
2
6
0
7
1
8
2
9
0
...... 40
1
0
1
1
41
1
1
2
2
42
1
2
3
0
43
1
0
1
1
44
1
1
2
2
45
1
2
3
0
46
1
0
1
1
.....
71
and one for divisibility by 4
10-digit
1-digit
2*10-digit
10-digit
1-digit
+ 1-digit 1
1
2
2
3
3
4
0
5
1
6
2
7
3
8
0
9
1
+ 1-digit
...... 30
3
0
6
2
31
3
1
7
3
32
3
2
8
0
33
3
0
9
1
34
3
1
6
2
35
3
2
7
3
36
3
0
8
0
.....
2*10-digit
72 If I were to teach this material once again, I might begin with an exploration92 of such patterns and then motivate my definition of modular arithmetic and discussion of its properties by, in essence, the question “Why?” I might begin with, for example, with “Is 1675 is divisible by 3, and, if so, why or why not?” Then I might note that this question is equivalent to the question 1675 ≡ ? mod 3 and suggest we begin by exploring the properties of this notation in the following: Now, it is clear that 4 ≡ 1 mod 3 and 8 ≡ 2 mod 3 A quick calculation should convince that 12 = 8 + 4 ≡ 1 + 2 mod 3 So let me conjecture93 that the modular relationship is additive. That is if a ≡ c mod 3 and b ≡ d mod 3 92 93
That is, explore, in effect, the remainders I give a proof of these conjectures in the book
73
then94 a + b = c + d mod 3 If modular notation is additive it is certainly multiplicative. That is if a ≡ c mod 3 then 2⋅a ≡ 2⋅c mod 3 For example, 4 ≡ 1 mod 3 and it is clear that 8 ≡ 2 mod 3
Now 10 ≡ 1 mod 3 so we know that 100 ≡ 10 mod 3
94
And, of course, this is true for any modulus.
74 However, 10 ≡ 1 mod 3
(A)
and since the modular notation is transitive and commutative95 100 ≡ 1 mod 3
(B)
Further, again taking advantage of the multiplicative property of the modular notation gives 1000 ≡ 1 mod 3
(C)
Thus, to summarize, I have using (A), (B), (C) 1⋅1000 ≡ 1⋅1 mod 3 6⋅100 ≡ 6⋅1 mod 3 7⋅10 = 7⋅1 mod 3 And, as the modular relationship is additive and transitive96, 1675 ≡ 1000 + 600 + 70 + 5 mod 3 1675 ≡ 1 + 6 + 7 + 5 mod 3 Hence
95
Transitivity implies that if a ≡ b mod 3 and b ≡ d mod 3 then a ≡ d mod 3. Commutativity implies that if a ≡ b mod 3 then b ≡ a mod 3 96 Now is a good time to refer back to the work these mathematics education students just did with the pattern worksheet.
75 1675 ≡ 1 + 0 + 0 + 2 mod 3 ≡ 3 mod 3 ≡ 0 mod 3
At this point, in the lesson I might ask my mathematics education students to show, for example, why97 the 4-divisibility pattern works, and depending on comfort level of the students I could also ask for proof of certain of the properties of modular arithmetic. Casting Out Nines This technique is reasonably accessible to elementary school students in the upper grades and may be seen by them as useful and intriguing. It is evident from the previous discussion, for example, that 1675 ≡ 1 + 6 + 7 + 5 mod 9 ≡ 1 mod 9 and this behavior of remainders of 9 can be explored by students using a table much like those I considered earlier
97
A reasonable discussion of casting out nines and elevens entails some discussion and analysis of these divisibility patterns
76 10-digit
1-digit
2*10-digit
10-digit
1-digit
+ 1-digit 1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
0
2*10-digit + 1-digit
...... 30
3
0
3
3
31
3
1
4
4
32
3
2
5
5
33
3
3
6
6
34
3
4
7
7
35
3
5
8
8
36
3
6
9
0
..... Assuming that students are comfortable98 with such behavior, it follows, for example, that
98
I am not mandating that students be, more generally, introduced to modular arithmetic, but that, with reasonable justification, they take advantage of some procedural rules.
77 2073 ≡ 2 + 0 + 7 + 3 mod 9 ≡ 3 mod 9 and, thus99, 1675 + 2073 = 3748 ≡ 3 + 7 + 4 + 8 mod 3 ≡ 4 mod 9 Which, I note, simply follows from the additive properties of the modular notation. Hence to check a complicated addition a student need only ▪
Add the digits of each addend and compute the corresponding remainder by division by 9 (that is, cast out all multiples of nine). In my example: ‘1’ and ‘3’ respectively
▪
Add these remainders while again casting out nines. In my example ‘4’
▪
Add the digits of the arithmetic sum of the addends while again casting out nines. In my example ‘4.’
▪
Compare the results of step 2 and 3. If they differ, an arithmetic mistake has occurred.
It is important to stress that if the results are the same100, an arithmetic mistake may still have occurred. In the pre-calculator days when casting out nines was more of a common practice among accountants a further check was made by, in effect, casting out elevens. Davydov Once Again Depending on the circumstances, an instructor in similar circumstances as myself, might find it worthwhile to return to Davydov’s discussion of the structural properties of length measurement. For instance, if A, B, and C are lengths A = B implies B = A 99
(that is, Commutativity101)
The sum of the remainders modulus nine equals the remainder of the sum modulus nine I give a simple example in the book
100
78
A = B and B = C implies A = B
(that is, Transitivity)
A < B and B < C implies A < C A = B implies A + C = B + C
(that is, Additivity)
A < B implies A + C < B + C A = B implies n⋅A = n⋅B
(n ≥ 0) (that is, Multiplicativity)
A < B implies n⋅A < n⋅B
(n ≥ 0)
That is, measurement of lengths and modular arithymetic are, to a degree, similar instances of more general mathematical structures. This takes on greater significance the more deeply one looks into patterns.
Augmented Problem Set 1.
Show the following are not primes: (a) 71; (b) 103; (b) 241
2.
Divide 233 by 16 using the Egyptian method
3.
Solve and show your work (a) 2316 ≡ ? (mod 3); (b) 34788 ≡ ? (mod 3)
4.
Solve and show your work (a) 23175 ≡ ? (mod 9); (b) 347886 ≡ ? (mod 9)
5.
Give a convincing argument that a number is divisible by 3 if the sum of its digits is divisible by 3.
6.
Give a convincing argument that a number is divisible by 9 if the sum of its digits is divisible by 9.
101
Note that the ‘0 then n > nm – d ≥ 0 Proof If r = 0 then nm - d = 0, hence I need to only consider the case when r > 0. I have substituting m-1 for q. d = (m-1) n + r As m-1 > 0
=
+
Thus
>
and d > (m-1)n or n > nm – d
117
However nm – d = n (q+1) –d = n –r > 0 So n > nm – d ≥ 0 Estimation by Unit Fractions
Let
be a fraction such hat 0
1 and d = nq + r for n > r >0 By Lemma III I can choose m1 such that 146
That is, my original fraction is a unit fraction
118
n > nm1 – d ≥ 0 I now consider the remainder
I set n1 = nm1-d d1 = dm1 and consider d1 = n1q1 + r1
If n1 = 1 or d1 = n1q1 (i.e. r = 0) I am done147. If not I can use Lemma III to find m2 such that n1 > n1m2 – d1 ≥ 0 That is, n > n1 > n1m2 – d1 ≥ 0
It is clear that I can continue this process if necessary by considering the remainder such that 147
That is, the remainder is a unit fraction.
119
n2 = n1m2 – d1 d2 = d1m2 and so on. However the integer sequence n1, n2...ns is monotonically decreasing n > n1 > n2 > ….> ns > 0 and hence must be finite. Augmented Problem Set (Unit Fractions) 1. Write a unit fraction expansion for 16/17 2. What is the smallest fraction that cannot be written with less than three unit fractions? How do you know? 3. Prove that
does not have a unique148 expansion in terms of unit fractions.
4. Find four different unit fractions that add to 1
Equivalent fractions
Just as I can represent a length of inch—I can represent it by equivalent to
148
inch by
inch—that is, within
inch +
inch—that is, within a
inch. We say, in this case, that
inch and mean by that the magnitude of a length of
Unique refers to the sum
inch is
inch is equal to the
120
magnitude of a length of
inch keeping in mind that measuring the length to within
different than measuring it in to within
is quite
.
Although this notation provides an efficient way of recording fractions—for example, seven
s is recorded149 as
— it tends to obscure the magnitude of the fraction. However, I can
with a little ingenuity use the idea of equivalence in estimation of such magnitudes. Consider, for example, the fraction
. It is obvious that
magnitudes are the same. How about smaller than
. How about
—is larger than
? Well
? Well a
is equivalent to is
is larger than a
; that is, their
less than
so it is
so clearly150
—that is,
.
Augmented Problem Set (Equivalent Fractions)
1.
Use equivalent fractions to estimate
2.
Use equivalent fractions to prove that
3.
Use equivalent fractions to estimate
149
Teachers tend to introduce fractions as
is n of the 150
The Number Line If one takes the unit interval as a unit, positive fractions151 can be represented on the number line as magnitudes that correspond to their distance from zero. For instance, the fraction is represented as being three fraction152
s from zero in the direction of increase, and in like manner the
is represented as being five
s from zero in the direction of increase. As this latter
fraction also corresponds to a distance of one unit and one-fourth of a unit, we often write153 it as 1
; i.e. 1 +
. All fractions with an absolute magnitude greater-than-or-equal-to 1can be
written as an improper or mixed number. The former notation is advantageous when engaging in multiplicative operations, the later is advantageous when engaging in additive operations. Note that, for example, while magnitude of
is, formally, not yet a fraction, it does have the
. Thus, it is evident, for example, that
cb (Hint: if two fractions have the same denominator you need only compare numerators)
A Final Comment A number of years ago I was doing some consulting in an urban school. It was near the beginning of the school year and I chanced to engage in conversation a young man going into 6th grade. He expressed enthusiasm for the learning of mathematics and seemed especially enthusiastic about learning about fractions. I chanced, a few months into the school year, to again engage this student in conversation. Unfortunately, he seemed now quite disenchanted with mathematics.
138 Children should learn to invert and multiply when dividing by fractions, but as a shortcut. Too often they are left with no conceptual understanding of fractions and their operations and, hence, no substantial problem solving skills entailing fractions. This does not bode well for mathematics to come which leans heavily on decimal fractions and real numbers.
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