156 Mathematics Teaching in the Middle School                                    ●   Vol. 15, No. 3, October 2009Copyright...
Building                         Numbers                         fromPr mes                         Jerry Burkhart        ...
Fig. 1 A listing of prime numbers upFigure their representative color block                                    to 20 and  ...
very important idea: Counting num-           Fig. 2 Building the counting numbers 2 through 12. The colored blocks help to...
Fig. 4 The numbers 1 through 50 with factors. The poster-board grid often remains on the classroom wall for reference afte...
appearances seem to be distributed           Fig. 5 The process of multiplying 88 by 75. Using the commutative and associa...
Fig. 6 Dividing 60 by 10 occurs by detaching blocks.                                                If we multiply numbers...
discussion of multiplication blocks,        Fig. 9 The factors of 210 can be found by detaching a single block or clusters...
Fig. 10 Multiples of a number are formed by attaching blocks to its diagram. In this       this number? Yes, they say, bec...
THE LCM                                       Fig. 12 When the least common multiple of two numbers is defined as “the sma...
using prime factorizations. This type       Fig. 14 Many strategies evolve for building the LCM of two numbers, if their b...
Fig. 15 Using the prime number blocks, rules emerge for finding the GCF and LCM.                    GCF(a, b) × LCM(a, b) ...
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Building Primes

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Building Primes

  1. 1. 156 Mathematics Teaching in the Middle School ● Vol. 15, No. 3, October 2009Copyright © 2009 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
  2. 2. Building Numbers fromPr mes Jerry Burkhart p Use building blocks to create a visual model for prime factorizations. Prime numbers are often described as the “building blocks” of natural numbers. This article will show how my students and I took this idea literally by using prime factorizations to build numbers with blocks. Many fascinating patterns and relationships emerge when a visual image of prime factorizations can be formed. This article will begin by exploring— 1. divisibility, 2. prime and composite numbers, and 3. properties of exponents. The article will conclude by investigating the relationship between— 1. greatest common factors and 2. least common multiples. Students can USING MULTIPLICATION TO BUILD COUNTING NUMBERS explore many When we want to understand how something works in the physical world, we often look at how it is constructed from simpler pieces. If we want to know about concepts of number the properties of molecules, we must understand that they are built from the ele- ments that we see listed in the periodic table. Elements can then be analyzed by theory, including looking at their atomic structures. Similarly, when we understand that a number is built from its prime factors, we need to look at its properties and its relation- the relationship ship to other numbers. For the last few years, I have used the activities described here over a two- between greatest week period in my sixth-grade classroom. Students come to sixth grade having been introduced to the basic definitions of prime and composite numbers and the common factorsDex Image/Photolibrary procedures for finding prime factorizations. I place students in groups of two or three and distribute a set of colored cen- and least common timeter cubes to each group; square “blocks” cut from colored paper or cardboard could also be used. The groups are told that each color represents a different multiples. number, although they do not know which particular number, and that placing Vol. 15, No. 3, October 2009 ● Mathematics Teaching in the Middle School 157
  3. 3. Fig. 1 A listing of prime numbers upFigure their representative color block to 20 and 1 a factor of 3. We need a new color of building block for the number 3, which Color Prime Number Suggested Quantity is red. Students place a red block next per Group to the number 3 on their page. white 2 20 Do we need a new type of building block to build 4? Students will see that although we could build 4 as 4 × 1, we red 3 10 may also use our known 2 blocks to express it as 2 × 2. Since we will never orange 5 5 introduce a new type of building block unless it is needed, we attach two white 2 blocks to represent 2 × 2. Students yellow 7 5 place this pair of white blocks next to the number 4 on the page. green 11 3 As groups understand the process, blue 3 they begin to work more indepen- 13 dently. The next number is 5. Since it cannot be built using the factors of purple 17 2 2 or 3, the number 5 must have its own building block, which is orange. The number 6 can be written as 2 × 3. brown 19 2 Students attach a white 2 block and a red 3 block. The number 7 requires black* 5 a new block, which is yellow. Attach- ing three white 2 blocks as 2 × 2 × 2 *Black blocks may be used to represents 8. represent any prime number greater than 19. Students continue building the re- maining counting numbers 2 through 12, as summarized in figure 2. Noteblocks together means the numbers How can we use multiplication to that the commutative and associativeare to be multiplied. make the number 2? Students usually properties of multiplication imply that Each group writes the numbers 2 state that 2 can be written as 2 × 1 a given collection of blocks representsthrough 12 in a column on the left only, disregarding the order of the a unique number, regardless of howside of a sheet of paper. The lesson factors. On one hand, they realize that they are ordered or grouped.begins with the class building each of we may use as many factors of 1 asthese numbers. As the activity pro- we like. On the other hand, we really The building blocks we have neededgresses and the students discover the do not need any of them. To keep the so far are 2, 3, 5, 7, and 11. What dotypes of building blocks needed, each process simple, we will ignore the fac- these numbers have in common? Stu-color will be assigned an appropri- tors of 1 and use a single white block dents often first point out that most ofate number so that we can build the to represent the number 2. To show the numbers are odd. However, theycounting numbers. (See fig. 1.) this, each group places a white block see that 2 is included although 9 is This article will use a question- next to the number 2 on their paper. not. Someone then comments that alland-answer format to mirror the con- five numbers are prime and that a newtent of a typical classroom discussion. How can we use multiplication to color of block will always be neededFor clarification, we begin by defining build the number 3? Students respond because, by definition, prime numbersthe counting (natural) numbers as that just as with the first number, 3 cannot be built from other factors.the set of positive integers: {1, 2, 3, 4, can be written as 3 × 1, but once again However, all composite numbers5, . . .}. Notice that this set does not the 1 is not needed. We are not able to so far have been built from existinginclude the number 0. use our white 2 block because 2 is not prime blocks. We then summarize this158 Mathematics Teaching in the Middle School ● Vol. 15, No. 3, October 2009
  4. 4. very important idea: Counting num- Fig. 2 Building the counting numbers 2 through 12. The colored blocks help to visual-bers are built by multiplication using ize the factors of composite numbers and identify prime numbers.prime numbers as building blocks. Natural Number Block Representation We have now built the counting (Prime Factorization)numbers 2 through 12. Is it possible if wekeep building each counting number in 2 2order that we will come to a number thatcannot be constructed? Students may 3 3note that this is impossible becausewhenever a number cannot be builtusing existing blocks, we introduce a 4 2 2new type of block to represent it. 5 5 At this point in the activity, thegroups continue building the count- 6 3ing numbers consecutively, beginning 2with the number 13, listing morenumbers, and placing the appropriate 7 7“building” next to each number. Tomake the next portion of the activity 8 22 2run more smoothly, they also sketcheach “building” as they create it. Although the students have been 9 3 3exposed briefly to the concept of aprime factorization, most of them do 10 5not immediately make the connection 2with this activity. I do not give themthe vocabulary or any algorithms. 11 11 Watching the strategies that thegroups develop is the overriding con- 12 3cern at this point. I monitor the groups 2 2as they work, giving clues if they arestuck and helping them identify andcorrect errors. If they run out of blocks Fig. 3 To build the number 72, divide prime numbers into the previous quotient andof a given color, they can disassemble collect the cubes from successful divisions until the quotient is 1.some of their previous constructionsand use those blocks, since they havealready sketched the diagrams. Once most groups have found atleast one successful strategy, the classshares their discoveries. Some groupsuse a variation of the one-block-at-a-time strategy, illustrated in figure 3.Others realize that they are findingprime factorizations and building fac-tor trees. (If so, I introduce the neces-sary vocabulary.) Still others write thenumber they are building as a productof two factors, look at their previ-ously built numbers, and attach copies Vol. 15, No. 3, October 2009 ● Mathematics Teaching in the Middle School 159
  5. 5. Fig. 4 The numbers 1 through 50 with factors. The poster-board grid often remains on the classroom wall for reference after theactivity is over, much like a periodic table in a chemistry class. 1 2 3 4 5 6 7 8 9 10 3 5 2 3 2 2 5 7 2 2 2 3 3 2 2 21 31 22 51 21 x 31 71 23 32 21 x 51 11 12 13 14 15 16 17 18 19 20 3 13 7 5 3 3 19 5 11 2 2 2 2 17 2 2 2 3 2 2 2 111 22 x 31 131 21 x 71 31 x 51 24 171 21 x 32 191 22 x 51 21 22 23 24 25 26 27 28 29 30 7 11 3 13 7 5 23 5 5 3 3 3 29 3 2 2 2 2 2 2 2 3 2 31 x 71 21 x 111 231 23 x 31 52 21 x 131 33 22 x 71 291 21 x 31 x 51 31 32 33 34 35 36 37 38 39 40 11 17 7 3 3 19 13 5 31 2 2 2 2 2 37 3 2 5 2 2 2 3 2 2 2 311 25 31 x 111 21 x 171 51 x 71 22 x 32 371 21 x 191 31 x 131 23 x 51 41 42 43 44 45 46 47 48 49 50 7 11 23 3 43 5 5 5 41 3 47 7 7 2 2 3 3 2 2 2 2 2 2 2 411 2 1 x 31 x 71 431 22 x 111 32 x 51 21 x 231 471 24 x 31 72 21 x 52of these two factors’ buildings. For could use as many or as few 1 blocks ing prime factorizations up to 100example, suppose they are building 36. as we like to build any counting num- among the groups, and then share theirThey recognize 36 as 9 × 4 and use ber. It is much simpler to simply avoid results with the class. If time is short,the factors of 9 and 4 that they have its use. If some students wonder how I may use a prepared grid and movealready built. Specifically, they just at- to build the number 1, I tell them that directly into the discussion. (Fig. 4 is atach the two white 2 blocks to make 4 we will address it soon. completed grid through 50. )and the two red 3 blocks to make 9. We chose to arrange the blocks in a PATTERNS AND STRUCTURE consistent manner for ease of reading. Why do you think that prime num- Students work in their groups to create Repeated factors appear in horizontalbers are so important? Students reply a building-block grid for the prime rows, with smaller factors appearing be-that primes are the numbers needed factorizations of all counting numbers low the larger factors. Now that we haveto build the counting numbers using from 2 through 100. (The number 1 is completed grids for reference, furthermultiplication. included in the grid but is left blank.) class discussion can begin. The completed grid will be used to Why do you think that 1 is not de- generate a class discussion in which Do you see any relationship betweenfined as a prime number? In fifth grade, students observe, analyze, and describe the size of a number and the size of itsstudents learned that a prime number patterns within and between prime building? Students should notice thathas exactly two factors: 1 and itself. factorizations of different numbers. the apparent size of a block diagramThe fact that 1 is not defined as prime Each group receives a large, num- has no predictable relationship to theis noted simply as a mathematical bered 10 × 10 grid on poster board; size of the number it represents. Forconvention. One goal of these ac- colored pencils or markers; and a example, the number 60 requires fourtivities is to help students develop a template for drawing squares. Since blocks to be built; 61, although larger,sense for why this seemingly arbitrary this activity can be time-consuming, is prime and requires only one block.choice may have been made. It would students may refer to the sketches of Although patterns are found in thebe confusing to include 1 as a prime the numbers they have already built. I distributions of particular colors ofnumber (a building block), because we divide the task of finding the remain- blocks, buildings of different sizes and160 Mathematics Teaching in the Middle School ● Vol. 15, No. 3, October 2009
  6. 6. appearances seem to be distributed Fig. 5 The process of multiplying 88 by 75. Using the commutative and associativealmost randomly throughout the grid. properties of multiplication, blocks can be rearranged to make the product easier to calculate with mental math. Can you see a way to predict the loca-tion of the next prime number by looking 11at a previous prime number? (Remember 5 5 5 5that the prime numbers are those that are 11 3 3built with exactly one block.) Studentsmay spend quite a bit of time looking 2 2 2 attach 2 2 2for patterns in the distribution of the 23 x 111 x 31 x 52 = 23 x 31 x 52 x 111prime numbers. They often think they 88 x 75 = 6600have made a discovery only to finda suspected pattern soon falls apart.They are often surprised to learn that,in spite of a long search, mathemati-cians have never found a simple way 11 5 5to predict the next prime. 2 2 2 3 11 Certain types of patterns can befound. For example, do you notice any- 11 5 5thing about the buildings of neighboring 5 5 3 3numbers (numbers that differ by 1) on 2 2 2 2 2 2your grid? The only color block that 10 x 10 x 66 = 6600the buildings of neighboring numbershave in common is black. Black blocksrepresent any prime number larger contain at least one red 3 block. No a grid posted in the classroom. It canthan 19. When two neighboring other numbers contain red blocks. be a wonderful resource for identify-numbers both contain a black block, Students realize that white blocks ing prime numbers, verifying multipli-these blocks always represent different are always two squares apart; red cation facts, doing mental arithmetic,prime factors. For example, although blocks, three squares apart; orange finding greatest common factors and46 and 47 each contain a black block, blocks, five squares apart, and so on. least common multiples, and so on. Itone black represents the factor 23, Blocks that represent the same factor resembles a periodic table of countingwhereas the next black stands for 47. are always more than one square apart. numbers. However, our periodic table Translating our observation from Many other interesting questions contains all counting numbers, notblock language to the language of can be explored using the grid: just the building blocks.mathematics, numbers that differ by1 have no common factor other than 1. How can we see that every mul- MULTIPLICATION, DIVISION,1. At first glance, this may seem quite tiple of 4 is also a multiple of 2? AND EXPONENTSsurprising. However, it can be easily 2. If a number is divisible by both The next step asks students to analyzeunderstood by looking more closely at 2 and 3, how can we see that it is the blocks for multiplication.even and odd numbers. also divisible by their product, 6? 3. If a number is divisible by both How can we use blocks to show the What do the buildings of all even 4 and 6, how can we see that it multiplication of composite numbers?numbers have in common? What about is not necessarily divisible by the Attaching individual blocks (primethe buildings of odd numbers? The build- product, 24? numbers) means that the two num-ings of even numbers always have at 4. If two numbers differ by n (if they bers are being multiplied; we doleast one white 2 block. Odd numbers are n squares apart on the grid), then the same for composite numbers.never contain any white blocks. what can we say about the possible Figure 5 illustrates 88 × 75 being common factors of the numbers? produced with building blocks. What can we say about the buildings After we have completed the Blocks can also be rearranged to aidof multiples of 3? These buildings all building-blocks activities, I often leave mental multiplication. Vol. 15, No. 3, October 2009 ● Mathematics Teaching in the Middle School 161
  7. 7. Fig. 6 Dividing 60 by 10 occurs by detaching blocks. If we multiply numbers by attach- ing blocks, how would we divide them? Students usually suggest that blocks 5 should be separated or detached. 5 detach 2 Avoid phrases that suggest subtrac- tion, such as take away. Figure 6 3 3 illustrates the number sentence 60 ÷ 10 = 6. We build the number 60, 2 2 2 detach a 5 and a 2 block, and leave a 3 and a 2 block. (22 x 31 x 51) / (21 x 51) 21 x 31 Does it matter whether we detach the 2 and 5 blocks together or one at a time? 60 / 10 = 6 Students can easily see that whether we detach the blocks one at a time, or both at once, the resulting collectionFig. 7 Exponents are a way of expressing the repetition of a number block. Attaching of blocks is the same. In other words,blocks of the number increases the exponent, in this case showing that 23 × 22 = 23+2. division by 10 is equivalent to division by 2 followed by division by 5. Figures 5 and 6 also show ex- 2 22 2 2 2 2 2 2 2 ponential notation for the block attach diagrams. Notice that each exponent 23 x 22 = 25 just counts the number of blocks of a given prime factor (color). The blocks can help students naturally visualize properties of exponents. Figure 7’sFig. 8 Detaching blocks and seeing the exponent decrease helps students realize that block diagram represents the number30 = 1 when there are no blocks left. sentence 2m × 2n = 2m+n. If all blocks are the same color, then the number 3 of blocks of that color obtained by attaching the buildings is the sum of the number of blocks in each building. 27 = 33 (3 “3” blocks) 3 3 3 In mathematical language, if the bases (detach 1 “3” block) are the same, the exponent of the (divide by 3) product is the sum of the exponents of 3 the factors: 9 = 32 (2 “3” blocks) 3 3 am × an = am+n (detach 1 “3” block) (divide by 3) Using the fact that we divide by 3 detaching blocks, students may also be able to predict the corresponding 3 = 31 (1 “3” block) 3 property for division of exponential (detach 1 “3” block) expressions: (divide by 3) am 1 = 30 (0 “3” blocks) No Blocks an = a m−n Susie : 2is in last parasome students aphy be confused Since before bibliogr may r by the addition and subtraction in our 2162 Mathematics Teaching in the Middle School ● Vol. 15, No. 3, October 2009
  8. 8. discussion of multiplication blocks, Fig. 9 The factors of 210 can be found by detaching a single block or clusters ofI make a point of saying that we are blocks.not adding or subtracting the blocks(the prime numbers themselves).Rather, we are adding or subtracting 7 7 7 7 7 7the number of blocks of each type (the 5 5 105 5 5 70 5 3 42exponents). 3 3 x 3 2 x 3 2 x 2 2 2 2 3 3 2 5 5 What will happen to the blocks if westart with a building showing an expo-nential expression, such as 33, then keep 7 7 7 7 7 35 7 7 21dividing by the base? Students realizethat the blocks will be detached one at 5 5 x 5 5 x 5 3 xa time until none are left. 3 3 30 3 3 3 5 2 2 2 2 6 2 2 10 What number will it representwhen no blocks remain? Students’ first 7response is usually 0, which is con- 7 5 15 7 5 3 210tradicted by figure 8. The number 27 5 x 5 3 3 3 xis 33, so we build it with 3 red blocks. 7 2Each time we divide by 3, the expo- 2 14 2nent drops by 1, going from 3 to 2 to 2 no blocks 11 to 0. At the same time, the quotientsgo from 27 to 9 to 3 to 1. Apparently,the number 1 is built by using no, or to visualize what is happening. All the part that remains. It is helpful tozero, blocks. Since exponents count composite numbers consist of two remember that the “empty” collectionthe number of blocks, this must mean or more blocks. Prime numbers are of blocks represents the number 1.that 30 = 1. represented by exactly one block. The Once students have seen this example, This action is related to the fact natural number 1 contains no blocks. I ask them to try a number with morethat any number multiplied by 1 is In mathematical language, composite factors, such as 210. Challenge themequal to itself. As an example, carry numbers are built from two or more to organize the task of finding fac-out the procedure above in reverse. prime factors; prime numbers are built tor pairs to ensure that they do notIn the first step of the process, we from one such factor; and the number double count or leave out any pairs, asbegin with 0 red blocks and attach 1 is built from none. The number 1 is shown in figure 9.1 red block, which is represented by unique in this respect. However, ratherthe number sentence 1 × 3 = 3. If no than being an uncomfortable exception Why do you think 210 has so manyblocks, or exponents of 0, were to to the pattern, it fits naturally into it. more factors than 12? Some studentsrepresent the number 0, then attach- may think at first that this is becauseing blocks to no blocks would repre- FACTORS AND MULTIPLES 210 is a larger number. I remind themsent multiplication by 0. The product Building blocks provide a striking way that prime numbers may be very large,would always be 0, regardless of the to visualize the factors of a number. yet have only two factors. I refer toblocks that we attached. Since we obtain factors by division, finding factors of 12 and 210, guid- Look more closely at the way we we form a factor of a number from ing them to understand that the totalcategorize the counting numbers. The its block diagram by detaching any number of factors depends on theFundamental Theorem of Arithme- number of blocks (including none number of prime factors as well astic states that every natural number or all of them). To be more precise, how many of them are distinct.except 1 can be expressed uniquely as a the factors in any block diagram areproduct of prime factors, disregarding exactly all possible subcollections Suppose we create a building to rep-their order. Excluding the number 1 of its blocks. Each time we detach resent some number, n, with four white 2may seem a little awkward. However, blocks, we actually create a pair of blocks, two red 3 blocks, two yellow 7 blocks,the blocks give us a compelling way factors: the part that we detached and and one blue 13 block. Is 56 a factor of Vol. 15, No. 3, October 2009 ● Mathematics Teaching in the Middle School 163
  9. 9. Fig. 10 Multiples of a number are formed by attaching blocks to its diagram. In this this number? Yes, they say, becausecase, multiples of 12 are found. 56 = 23 × 7, and the building contains three white 2 blocks and a yellow 7 3 3 12 x 1 = 12 (attach no blocks) block. 2 2 2 2 Is 45 a factor? No. Since 45 = 2 32 × 5, students realize that an orange 3 3 12 x 2 = 24 (attach a “2” block) 5 block would be needed. 2 2 2 22 3 Is 32 a factor? Students say no. 3 3 3 12 x 3 = 36 (attach a “3” block) Since 32 = 25, we need five white 2 2 2 2 2 blocks, but we have only four white blocks. 3 2 2 3 We can answer these questions (attach 2 “2” blocks) 2 2 2 22 2 12 x 4 = 48 about factors without knowing the value of the number from merely see- 5 5 ing its prime factorization, which in 3 3 12 x 5 = 60 (attach a “5” block) this case is 91,728. After this activity, 2 2 2 2 students may want to calculate the value of building this bulky number, then divide to verify the responses toFig. 11 Finding the greatest common factor can be found easily when defined as “the the questions about this number.largest collection of blocks that is contained in both block diagrams” 54 each number. Building the Greatest Common Factor of 36 and for Next, the class uses blocks to build multiples. Figure 10 shows how to create five multiples of 12 by attach- 3 3 3 3 3 ing first no blocks, then a 2 block, 2 2 2 a 3 block, two 2 blocks (for the 36 54 number 4), and finally a 5 block to the original block diagram for 12. Common factors of 36 and 54 are collections of blocks The block diagram for each multiple that are contained in both of the above block diagrams: contains the original building. the “empty” 1 collection of blocks THE GCF The building-blocks model is an 2 2 especially powerful tool for discover- ing and describing the mathemati- 3 3 cal relationships inherent in greatest common factors (GCF) and least 3 6 common multiples (LCM). 2 Since factors are formed as subcol- lections of the blocks in a building, a 3 3 9 common factor of two numbers is any subcollection that the two buildings 3 3 18 have in common. Once all common 2 factors are identified, the greatest com- mon factor is found to be the unique The above collection containing the most blocks is the Greatest Common Factor of 36 and 54: such collection containing the most blocks. In short, the GCF is repre- 3 3 18 sented as “the largest collection of 2 blocks that is contained in both block diagrams.” (See fig. 11.)164 Mathematics Teaching in the Middle School ● Vol. 15, No. 3, October 2009
  10. 10. THE LCM Fig. 12 When the least common multiple of two numbers is defined as “the smallestSince we form multiples by attaching collection of blocks that contains both block diagrams,” building the LCM is a matter ofblocks to the original building, every collecting blocks, as shown for the examples of 6 and 15.multiple of a number will containthe original building. Thus, a com-mon multiple of two numbers willbe represented by any collection of 3 5blocks that contains both of the given 2 3block diagrams. To determine the leastcommon multiple, locate the smallest 6 15such collection. (See fig. 12.) Remov- Common multiples of 6 and 15 must containing any block from the LCM would both of the above block diagrams. For example:result in a building that is no longer amultiple of both numbers. Thus, theLCM is represented as “the smallest 7collection of blocks that contains both 5 5 5 5 5block diagrams.” Figure 13 gives an 3 3 3 3 3overview of block representations ofthe GCF and LCM. The statements 2 2 2 2 2differ by only a few key words. 90 300 30 210 Building least common mul-tiples from blocks is generally more The collection of this type containing the fewest possible blocks ischallenging than forming greatest the Least Common Multiple of 6 and 15.common factors. I place studentsin groups, give them a collection of 5blocks, and ask them to develop a 3 30strategy for finding the LCM of a pairof numbers such as 1848 and 3276. 2(If time is short, I will tell them inadvance how to build the two given If any block is removed from this building, it will no longer be a common multiple.numbers.) Some groups may needsome hints to get started. The fol-lowing discussion gives some idea ofthe different ways in which students Fig. 13 Using blocks to represent the greatest common factor and the least commonapproach the problem. multiple of numbers M and N Did anyone begin with one building Factor of N Multiple of Nand then think of which blocks to attach toit? Students use a building and attach A collection of blocks A collection of blocks contained in the block diagram containing the block diagram ofjust enough blocks to ensure that the of N Nother building is contained as well.These additional blocks will be con- Common Factor of M and N Common Multiple of M and Ntained in the second number but “miss- A collection of blocks A collection of blocksing” from the first. (See fig. 14, method contained in the block containing the block diagrams diagrams both of M and N of both M and N1.) In my experience, this is the mostcommon strategy that students adopt. Greatest Common Factor Least Common Multiple When students imagine how diffi- of M and N of M and Ncult this computation would have been The largest collection of blocks The smallest collection ofhad they made long lists of multiples contained in the block blocks containing the block diagrams of both M and N diagrams of both M and Nand looked for common numbers, theyappreciate the advantages of Vol. 15, No. 3, October 2009 ● Mathematics Teaching in the Middle School 165
  11. 11. using prime factorizations. This type Fig. 14 Many strategies evolve for building the LCM of two numbers, if their blockof strategy is often suggested to algebra diagram is known. This example demonstrates finding the LCM of 1848 and 3276.students who are trying to find leastcommon multiples of the denomina- 11 13tors of algebraic fractions. We cannot 7 7generally make ordered lists of mul- 3 3 3tiples of algebraic expressions. 2 2 2 2 2 1848 3276 Did anyone attach the original Method 1numbers, then eliminate extra blocks? Begin with either building 13 The result 13Some groups think of attaching both is the 11 3 least 11original buildings, since this will en- commonsure that both buildings are contained 7 Attach just the blocks needed so that the block diagram for multiple 7in the result. When they look more 3 the other number (3276) is 3 3closely, they see that some blocks 2 2 2 also contained 2 2 2 1848 72072contained in both buildings are not Method 2needed. They detach these blocks to Attach the remaining blocks to the Begin with either building other building The result 13obtain the smallest collection con- 7 13 is the 11taining both buildings. (See fig. 14, 11 least 11 3 2 7 common 7method 2.) Some students may notice 7 2 2 multiplethat these duplicate blocks represent 3 3 3 3 3 Detach the 2 2 2 2 2the GCF of the two numbers. 2 2 2 GCF of the two numbers 3276 72072 1848RELATING THE LCM Method 3 Select one color (factor) at a time. Choose the larger of the two collections of blocks.AND THE GCF from 1848 from 3276 the larger collection attach to form the LCMIn my experience, most students blue none 13 13tend to remember this method the 13best and use it most often in the green 11 none 11 11long term, possibly because it can be yellow 7 7 7 7interpreted as a simple computation.The details of the computation will 3 3 red 3 3 3 3 3 2 2 2depend on the order in which stu-dents manipulate the blocks. Detach- white 2 2 2 2 2 2 2 2 72072ing the GCF from one building andthen attaching the resulting collec-tion of blocks to the other building of 24 and 18 is 6, the LCM may be become LCM(a, b) = ab; the numbersis equal to dividing one number by calculated as have no common factor other than 1,the GCF and multiplying the result and their buildings have no blocks inby the other number. In algebraic 24 ÷ 6 × 18 = 4 × 18 = 72 common. Such pairs of numbers arelanguage, we may write this as called relatively prime. or LCM(a, b) = a ÷ GCF(a, b) × b. Did any groups focus on one color 18 × 24 ÷ 6 = 432 ÷ 6 = 72. at a time and decide how many of eachAttaching the buildings before de- color are needed to form the least commontaching the blocks that they have in You may have noticed that the least multiple? Students rarely discovercommon gives us common multiple of two numbers can this strategy, but I usually guide them sometimes be found by multiplying them. through it. LCM(a, b) = a × b ÷ GCF(a, b). When can this be done? I encourage students to look closely at the formu- If the first number contains fourStudents may like to see some simple las to help them answer this question. white blocks and the second numberexamples. For instance, since the GCF When GCF(a, b) = 1, the formulas contains six white blocks, how many166 Mathematics Teaching in the Middle School ● Vol. 15, No. 3, October 2009
  12. 12. Fig. 15 Using the prime number blocks, rules emerge for finding the GCF and LCM. GCF(a, b) × LCM(a, b) = ab.Specifically, for each color, the smaller number of blocks from a and b goes to the GCFand the larger number of blocks goes to the LCM, as in the examples shown for 1008 This formula is equal to those foundand 9450. from the second strategy. a GCF(a,b) SUMMARY 7 7 Concrete representations of mathe- matical structures can engage stu- 33 33 dents’ interest and help them visualize 2 2 2 2 2 and internalize challenging concepts. 1008 126 By using building blocks to represent prime factorizations, my students have gained a deeper appreciation for b LCM(a,b) the structure of the natural numbers. 7 7 Along the way, they have discovered 5 5 5 5 many interesting patterns, relation- ships, and procedures. 333 333 Some readers may be inter- 2 2 2 2 2 ested in thinking about how the 9450 75600 building-blocks model might be extended to other concepts, such as Attach GCF(a,b) negative exponents and reciprocals, Attach a and b and LCM(a,b) fractional exponents and radicals, and logarithms. Blocks may also be used to help visualize classic proofs ab = GCF(a,b) x LCM(a,b) such as the irrationality of 12 and 7 7 7 7 the infinity of prime numbers. 5 5 5 5 Ideas and related problems, activi- 3 33 33 3 33 33 ties, and games are found at 2 2 2 2 2 2 2 2 22 http://themathroom.org. 9,525,600 9,525,600 BIBLIOGRAPHY Robbins, Christina, and Thomasenia Lottwhite blocks will their LCM contain? tioning, students discover that the Adams. “Get ‘Primed’ to the BasicAfter some thought, students will see collection of blocks that the build- Building Blocks of Numbers.” Math-that it must contain six white blocks. ings have in common (the GCF) is ematics Teaching in the Middle School 13All six are needed to ensure that the determined by the smaller of the two (September 2007): 122−27.second number is contained in the collections for each factor. Therefore, Zazkis, Rina, and Peter Liljedahl.LCM, but the additional four blocks if we have block diagrams for two “Understanding Primes: The Role offrom the first number are not neces- numbers, a and b, we may simultane- Representation.” Journal for Research insary since they are already included. ously form the GCF and the LCM. Mathematics Education 35 (May 2004):In general, for each color (prime We proceed, one color at a time, by 164−86.factor), the LCM will always contain attaching the larger collection tothe larger of the two collections of the LCM and the smaller collection Jerry Burkhart, jburkh1@blocks. For prime factorizations writ- to the GCF. Since this process uses isd77.k12.mn.us, teachesten in exponential form, this means each block in the two buildings ex- sixth-grade mathematicsthat, for each base, we choose the actly once, we can see that attaching in the Mankato Area Pub-larger of the two exponents. (See the GCF and the LCM will produce lic Schools in Minnesota.fig. 14, method 3.) the same result as attaching the two He is interested in finding new ways to We can take this idea one step original numbers. (See fig. 15.) The use mathematical models to help studentsfurther. Using a similar line of ques- formula is make sense of challenging concepts. Vol. 15, No. 3, October 2009 ● Mathematics Teaching in the Middle School 167

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