building primes

156 MatheMatics teaching in the Middle school ● Vol. 15, No. 3, October 2009

Copyright © 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.

Vol. 15, No. 3, October 2009 ● MatheMatics teaching in the Middle school 157

pPrime 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.

Using MUltiPlication to BUild coUnting nUMBeRs 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 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 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-ship to other numbers.

For the last few years, I have used the activities described here over a two-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 procedures for finding prime factorizations.

I place students in groups of two or three and distribute a set of colored cen-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 number, although they do not know which particular number, and that placing

fromPr mesnumbersBuilding

Use building blocks to create a visual model for prime factorizations. Students can explore many concepts of number theory, including the relationship between greatest common factors and least common multiples.

Jerry Burkhart

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blocks together means the numbers are to be multiplied.

Each group writes the numbers 2 through 12 in a column on the left side of a sheet of paper. The lesson begins with the class building each of these numbers. As the activity pro-gresses and the students discover the types of building blocks needed, each color will be assigned an appropri-ate number so that we can build the counting numbers. (See fig. 1.)

This article will use a question-and-answer format to mirror the con-tent of a typical classroom discussion. For clarification, we begin by defining the counting (natural) numbers as the set of positive integers: {1, 2, 3, 4, 5, . . .}. Notice that this set does not include the number 0.

a factor of 3. We need a new color of building block for the number 3, which is red. Students place a red block next to the number 3 on their page.

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 may also use our known 2 blocks to express it as 2 × 2. Since we will never introduce a new type of building block unless it is needed, we attach two white 2 blocks to represent 2 × 2. Students place this pair of white blocks next to the number 4 on the page.

As groups understand the process, they begin to work more indepen-dently. The next number is 5. Since it cannot be built using the factors of 2 or 3, the number 5 must have its own building block, which is orange. The number 6 can be written as 2 × 3. Students attach a white 2 block and a red 3 block. The number 7 requires a new block, which is yellow. Attach-ing three white 2 blocks as 2 × 2 × 2 represents 8.

Students continue building the re-maining counting numbers 2 through 12, as summarized in figure 2. Note that the commutative and associative properties of multiplication imply that a given collection of blocks represents a unique number, regardless of how they are ordered or grouped.

The building blocks we have needed so far are 2, 3, 5, 7, and 11. What do these numbers have in common? Stu-dents often first point out that most of the numbers are odd. However, they see that 2 is included although 9 is not. Someone then comments that all five numbers are prime and that a new color of block will always be needed because, by definition, prime numbers cannot be built from other factors. However, all composite numbers so far have been built from existing prime blocks. We then summarize this

Figure 1

Color Prime Number Suggested Quantity per Group

white 2red

orange

yellow

green

blue

purple

brown

black*

3

5

7

11

13

17

19

20

10

5

5

3

3

2

2

5

*Black blocks may be used torepresent any prime number greater than 19.

Fig. 1 a listing of prime numbers up to 20 and their representative color block

How can we use multiplication to make the number 2? Students usually state that 2 can be written as 2 × 1 only, disregarding the order of the factors. On one hand, they realize that we may use as many factors of 1 as we like. On the other hand, we really do not need any of them. To keep the process simple, we will ignore the fac-tors of 1 and use a single white block to represent the number 2. To show this, each group places a white block next to the number 2 on their paper.

How can we use multiplication to build the number 3? Students respond that just as with the first number, 3 can be written as 3 × 1, but once again the 1 is not needed. We are not able to use our white 2 block because 2 is not


very important idea: Counting num-bers are built by multiplication using prime numbers as building blocks.

We have now built the counting numbers 2 through 12. Is it possible if we keep building each counting number in order that we will come to a number that cannot be constructed? Students may note that this is impossible because whenever a number cannot be built using existing blocks, we introduce a new type of block to represent it.

At this point in the activity, the groups continue building the count-ing numbers consecutively, beginning with the number 13, listing more numbers, and placing the appropriate “building” next to each number. To make the next portion of the activity run more smoothly, they also sketch each “building” as they create it.

Although the students have been exposed briefly to the concept of a prime factorization, most of them do not immediately make the connection with this activity. I do not give them the vocabulary or any algorithms.

Watching the strategies that the groups develop is the overriding con-cern at this point. I monitor the groups as they work, giving clues if they are stuck and helping them identify and correct errors. If they run out of blocks of a given color, they can disassemble some of their previous constructions and use those blocks, since they have already sketched the diagrams.

Once most groups have found at least one successful strategy, the class shares their discoveries. Some groups use a variation of the one-block-at-a-time strategy, illustrated in figure 3. Others realize that they are finding prime factorizations and building fac-tor trees. (If so, I introduce the neces-sary vocabulary.) Still others write the number they are building as a product of two factors, look at their previ-ously built numbers, and attach copies

Natural Number Block Representation(Prime Factorization)

2

3

4

5

6

7

8

9

10

11

12

2

3

2

2 2

5

3

7

2 2 2

3 3

5211

223

Fig. 2 Building the counting numbers 2 through 12. The colored blocks help to visual-ize the factors of composite numbers and identify prime numbers.

Fig. 3 To build the number 72, divide prime numbers into the previous quotient and collect the cubes from successful divisions until the quotient is 1.


of these two factors’ buildings. For example, suppose they are building 36. They recognize 36 as 9 × 4 and use the factors of 9 and 4 that they have already built. Specifically, they just at-tach the two white 2 blocks to make 4 and the two red 3 blocks to make 9.

Why do you think that prime num-bers are so important? Students reply that primes are the numbers needed to build the counting numbers using multiplication.

Why do you think that 1 is not de-fined as a prime number? In fifth grade, students learned that a prime number has exactly two factors: 1 and itself. The fact that 1 is not defined as prime is noted simply as a mathematical convention. One goal of these ac-tivities is to help students develop a sense for why this seemingly arbitrary choice may have been made. It would be confusing to include 1 as a prime number (a building block), because we

could use as many or as few 1 blocks as we like to build any counting num-ber. It is much simpler to simply avoid its use. If some students wonder how to build the number 1, I tell them that we will address it soon.

PatteRns and stRUctUReStudents work in their groups to create a building-block grid for the prime factorizations of all counting numbers from 2 through 100. (The number 1 is included in the grid but is left blank.) The completed grid will be used to generate a class discussion in which students observe, analyze, and describe patterns within and between prime factorizations of different numbers.

Each group receives a large, num-bered 10 × 10 grid on poster board; colored pencils or markers; and a template for drawing squares. Since this activity can be time-consuming, students may refer to the sketches of the numbers they have already built. I divide the task of finding the remain-

ing prime factorizations up to 100 among the groups, and then share their results with the class. If time is short, I may use a prepared grid and move directly into the discussion. (Fig. 4 is a completed grid through 50. )

We chose to arrange the blocks in a consistent manner for ease of reading. Repeated factors appear in horizontal rows, with smaller factors appearing be-low the larger factors. Now that we have completed grids for reference, further class discussion can begin.

Do you see any relationship between the size of a number and the size of its building? Students should notice that the apparent size of a block diagram has no predictable relationship to the size of the number it represents. For example, the number 60 requires four blocks to be built; 61, although larger, is prime and requires only one block. Although patterns are found in the distributions of particular colors of blocks, buildings of different sizes and

3

2 2 532 7 2 2 2 3 32 2

5

1132 2

1327

35 2 2 2 2 17 3

23 19

2 25

37

211 23

2 2 23

55 213

3 3 27

229

3

35

2

31 2 2 2 2 2 311

217

57 3

232

372

193

132 2 25

41 32

743

2 211

3 35

223

47 2 2 2 23

77 552

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

21 31 22 51 21 x 31 71 23 32 21 x 51

111 22 x 31 131 21 x 71 31 x 51 24 171 21 x 32 191 22 x 51

31 x 71 21 x 111 231 23 x 31 52 21 x 131 33 22 x 71 291 21 x 31 x 51

311 25 31 x 111 21 x 171 51 x 71 22 x 32 371 21 x 191 31 x 131 23 x 51

411 21 x 31 x 71 431 22 x 111 32 x 51 21 x 231 471 24 x 31 72 21 x 52

Fig. 4 the numbers 1 through 50 with factors. the poster-board grid often remains on the classroom wall for reference after the activity is over, much like a periodic table in a chemistry class.


appearances seem to be distributed almost randomly throughout the grid.

Can you see a way to predict the loca-tion of the next prime number by looking at a previous prime number? (Remember that the prime numbers are those that are built with exactly one block.) Students may spend quite a bit of time looking for patterns in the distribution of the prime numbers. They often think they have made a discovery only to find a 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 to predict the next prime.

Certain types of patterns can be found. For example, do you notice any-thing about the buildings of neighboring numbers (numbers that differ by 1) on your grid? The only color block that the buildings of neighboring numbers have in common is black. Black blocks represent any prime number larger than 19. When two neighboring numbers both contain a black block, these blocks always represent different prime factors. For example, although 46 and 47 each contain a black block, one black represents the factor 23, whereas the next black stands for 47.

Translating our observation from block language to the language of mathematics, numbers that differ by 1 have no common factor other than 1. At first glance, this may seem quite surprising. However, it can be easily understood by looking more closely at even and odd numbers.

What do the buildings of all even numbers have in common? What about the buildings of odd numbers? The build-ings of even numbers always have at least one white 2 block. Odd numbers never contain any white blocks.

What can we say about the buildings of multiples of 3? These buildings all

contain at least one red 3 block. No other numbers contain red blocks.

Students realize that white blocks are always two squares apart; red blocks, three squares apart; orange blocks, five squares apart, and so on. Blocks that represent the same factor are always more than one square apart.

Many other interesting questions can be explored using the grid:

1. How can we see that every mul-tiple of 4 is also a multiple of 2?

2. If a number is divisible by both 2 and 3, how can we see that it is also divisible by their product, 6?

3. If a number is divisible by both 4 and 6, how can we see that it is not necessarily divisible by the product, 24?

4. If two numbers differ by n (if they are n squares apart on the grid), then what can we say about the possible common factors of the numbers?After we have completed the

building-blocks activities, I often leave

a grid posted in the classroom. It can be a wonderful resource for identify-ing prime numbers, verifying multipli-cation facts, doing mental arithmetic, finding greatest common factors and least common multiples, and so on. It resembles a periodic table of counting numbers. However, our periodic table contains all counting numbers, not just the building blocks.

MUltiPlication, diVision, and eXPonentsThe next step asks students to analyze the blocks for multiplication.

How can we use blocks to show the multiplication of composite numbers? Attaching individual blocks (prime numbers) means that the two num-bers are being multiplied; we do the same for composite numbers. Figure 5 illustrates 88 × 75 being produced with building blocks. Blocks can also be rearranged to aid mental multiplication.

2 attach

88 75x = 6600

23 x 111 x 31 x 52 = 23 x 31 x 52 x 111

2211 3

5 5

2 22

11

35 5

2 2211

35 5

25

25

2311

10 x 10 x 66 = 6600

2 22

11

35 5

Fig. 5 the process of multiplying 88 by 75. Using the commutative and associative properties of multiplication, blocks can be rearranged to make the product easier to calculate with mental math.


If we multiply numbers by attach-ing blocks, how would we divide them? Students usually suggest that blocks should be separated or detached. Avoid phrases that suggest subtrac-tion, such as take away. Figure 6 illustrates the number sentence 60 ÷ 10 = 6. We build the number 60, detach a 5 and a 2 block, and leave a 3 and a 2 block.

Does it matter whether we detach the 2 and 5 blocks together or one at a time? Students can easily see that whether we detach the blocks one at a time, or both at once, the resulting collection of blocks is the same. In other words, division by 10 is equivalent to division by 2 followed by division by 5.

Figures 5 and 6 also show ex-ponential notation for the block diagrams. Notice that each exponent just counts the number of blocks of a given prime factor (color). The blocks can help students naturally visualize properties of exponents. Figure 7’s block diagram represents the number sentence 2m × 2n = 2m+n. If all blocks are the same color, then the number of blocks of that color obtained by attaching the buildings is the sum of the number of blocks in each building. In mathematical language, if the bases are the same, the exponent of the product is the sum of the exponents of the factors:

am × an = am+n

Using the fact that we divide by detaching blocks, students may also be able to predict the corresponding property for division of exponential expressions:

aa

a

Susie

m

nm n= −

: 2is in last para before bibliogrraphy

2

Since some students may be confused by the addition and subtraction in our

22 2 22attach

222 22

23 x 22 = 25

Fig. 7 exponents are a way of expressing the repetition of a number block. attaching blocks of the number increases the exponent, in this case showing that 23 × 22 = 23+2.

27 = 33 333

3 3

3

9 = 32

3 = 31

1 = 30

(3 “3” blocks)

(2 “3” blocks)

(1 “3” block)

(0 “3” blocks)

(divide by 3)

(divide by 3)

(divide by 3)

(detach 1 “3” block)



No Blocks

3

3

3

Fig. 8 Detaching blocks and seeing the exponent decrease helps students realize that 30 = 1 when there are no blocks left.

2 235

52

detach

32

60 / 10 = 6

(22 x 31 x 51) / (21 x 51) 21 x 31

Fig. 6 Dividing 60 by 10 occurs by detaching blocks.


357

2 2357

105

x

2

357

2 3257

70

x

3

357

2 5

32

742

x

5

357

2

7

35

2

7

x

30 357

2357

2

35

x

6

357

2

7

325

21

x

10

357

2

35

72

15

x

14357

2

357

2

210

x

1no blocks

Fig. 9 the factors of 210 can be found by detaching a single block or clusters of blocks.

discussion of multiplication blocks, I make a point of saying that we are not adding or subtracting the blocks (the prime numbers themselves). Rather, we are adding or subtracting the number of blocks of each type (the exponents).

What will happen to the blocks if we start with a building showing an expo-nential expression, such as 33, then keep dividing by the base? Students realize that the blocks will be detached one at a time until none are left.

What number will it represent when no blocks remain? Students’ first response is usually 0, which is con-tradicted by figure 8. The number 27 is 33, so we build it with 3 red blocks. Each time we divide by 3, the expo-nent drops by 1, going from 3 to 2 to 1 to 0. At the same time, the quotients go from 27 to 9 to 3 to 1. Apparently, the number 1 is built by using no, or zero, blocks. Since exponents count the number of blocks, this must mean that 30 = 1.

This action is related to the fact that any number multiplied by 1 is equal to itself. As an example, carry out the procedure above in reverse. In the first step of the process, we begin with 0 red blocks and attach 1 red block, which is represented by the number sentence 1 × 3 = 3. If no blocks, or exponents of 0, were to represent the number 0, then attach-ing blocks to no blocks would repre-sent multiplication by 0. The product would always be 0, regardless of the blocks that we attached.

Look more closely at the way we categorize the counting numbers. The Fundamental Theorem of Arithme-tic states that every natural number except 1 can be expressed uniquely as a product of prime factors, disregarding their order. Excluding the number 1 may seem a little awkward. However, the blocks give us a compelling way

to visualize what is happening. All composite numbers consist of two or more blocks. Prime numbers are represented by exactly one block. The natural number 1 contains no blocks. In mathematical language, composite numbers are built from two or more prime factors; prime numbers are built from one such factor; and the number 1 is built from none. The number 1 is unique in this respect. However, rather than being an uncomfortable exception to the pattern, it fits naturally into it.

FactoRs and MUltiPlesBuilding blocks provide a striking way to visualize the factors of a number. Since we obtain factors by division, we form a factor of a number from its block diagram by detaching any number of blocks (including none or all of them). To be more precise, the factors in any block diagram are exactly all possible subcollections of its blocks. Each time we detach blocks, we actually create a pair of factors: the part that we detached and

the part that remains. It is helpful to remember that the “empty” collection of blocks represents the number 1. Once students have seen this example, I ask them to try a number with more factors, such as 210. Challenge them to organize the task of finding fac-tor pairs to ensure that they do not double count or leave out any pairs, as shown in figure 9.

Why do you think 210 has so many more factors than 12? Some students may think at first that this is because 210 is a larger number. I remind them that prime numbers may be very large, yet have only two factors. I refer to finding factors of 12 and 210, guid-ing them to understand that the total number of factors depends on the number of prime factors as well as how many of them are distinct.

Suppose we create a building to rep-resent some number, n, with four white 2 blocks, two red 3 blocks, two yellow 7 blocks, and one blue 13 block. Is 56 a factor of


32 2

32 2

12 x 1 = 12 (attach no blocks)

32 2

2 32 2 2

12 x 2 = 24 (attach a “2” block)

32 2

332 2

3 12 x 3 = 36 (attach a “3” block)

32 2

2 2 32 2 2 2 12 x 4 = 48 (attach 2 “2” blocks)

32 2

532 2

512 x 5 = 60 (attach a “5” block)

Fig. 10 multiples of a number are formed by attaching blocks to its diagram. In this case, multiples of 12 are found.

this number? Yes, they say, because 56 = 23 × 7, and the building contains three white 2 blocks and a yellow 7 block.

Is 45 a factor? No. Since 45 = 32 × 5, students realize that an orange 5 block would be needed.

Is 32 a factor? Students say no. Since 32 = 25, we need five white 2 blocks, but we have only four white blocks.

We can answer these questions about factors without knowing the value of the number from merely see-ing its prime factorization, which in this case is 91,728. After this activity, students may want to calculate the value of building this bulky number, then divide to verify the responses to the questions about this number.

Next, the class uses blocks to build multiples. Figure 10 shows how to create five multiples of 12 by attach-ing first no blocks, then a 2 block, a 3 block, two 2 blocks (for the number 4), and finally a 5 block to the original block diagram for 12. The block diagram for each multiple contains the original building.

the gcF The building-blocks model is an especially powerful tool for discover-ing and describing the mathemati-cal relationships inherent in greatest common factors (GCF) and least common multiples (LCM).

Since factors are formed as subcol-lections of the blocks in a building, a common factor of two numbers is any subcollection that the two buildings have in common. Once all common factors are identified, the greatest com-mon factor is found to be the unique such collection containing the most blocks. In short, the GCF is repre-sented as “the largest collection of blocks that is contained in both block diagrams.” (See fig. 11.)

Building the Greatest Common Factor of 36 and 54

32

32

3 3 32

36 54

Common factors of 36 and 54 are collections of blocksthat are contained in both of the above block diagrams:

2

3

32

3 3

3 32

the “empty” collection of blocks

1

2

3

6

9

18

The above collection containing the most blocks isthe Greatest Common Factor of 36 and 54:

3 32

18

Fig. 11 Finding the greatest common factor can be found easily when defined as “the largest collection of blocks that is contained in both block diagrams” for each number.


the lcMSince we form multiples by attaching blocks to the original building, every multiple of a number will contain the original building. Thus, a com-mon multiple of two numbers will be represented by any collection of blocks that contains both of the given block diagrams. To determine the least common multiple, locate the smallest such collection. (See fig. 12.) Remov-ing any block from the LCM would result in a building that is no longer a multiple of both numbers. Thus, the LCM is represented as “the smallest collection of blocks that contains both block diagrams.” Figure 13 gives an overview of block representations of the GCF and LCM. The statements differ by only a few key words.

Building least common mul-tiples from blocks is generally more challenging than forming greatest common factors. I place students in groups, give them a collection of blocks, and ask them to develop a strategy for finding the LCM of a pair of numbers such as 1848 and 3276. (If time is short, I will tell them in advance how to build the two given numbers.) Some groups may need some hints to get started. The fol-lowing discussion gives some idea of the different ways in which students approach the problem.

Did anyone begin with one building and then think of which blocks to attach to it? Students use a building and attach just enough blocks to ensure that the other building is contained as well. These additional blocks will be con-tained in the second number but “miss-ing” from the first. (See fig. 14, method 1.) In my experience, this is the most common strategy that students adopt.

When students imagine how diffi-cult this computation would have been had they made long lists of multiples and looked for common numbers, they appreciate the advantages of

32 3

5

6 15Common multiples of 6 and 15 must contain

both of the above block diagrams.For example:

32

35

32

5

2

532

532

57

90 300 30 210The collection of this type containing the fewest possible blocks is

the Least Common Multiple of 6 and 15.

32

530

If any block is removed from this building, it will no longer be a common multiple.

Fig. 12 When the least common multiple of two numbers is defined as “the smallest collection of blocks that contains both block diagrams,” building the lCm is a matter of collecting blocks, as shown for the examples of 6 and 15.

NfoelpitluMNforotcaF

Common Factor of M and N Common Multiple of M and N

Greatest Common Factor of M and N

Least Common Multipleof M and N

A collection of blockscontained in the block diagramof N

A collection of blockscontaining the block diagram ofN

A collection of blockscontained in the blockdiagrams both of M and N

A collection of blockscontaining the block diagramsof both M and N

The largest collection of blocks contained in the blockdiagrams of both M and N

The smallest collection of blocks containing the blockdiagrams of both M and N

Fig. 13 Using blocks to represent the greatest common factor and the least common multiple of numbers M and N


using prime factorizations. This type of strategy is often suggested to algebra students who are trying to find least common multiples of the denomina-tors of algebraic fractions. We cannot generally make ordered lists of mul-tiples of algebraic expressions.

Did anyone attach the original numbers, then eliminate extra blocks? Some groups think of attaching both original buildings, since this will en-sure that both buildings are contained in the result. When they look more closely, they see that some blocks contained in both buildings are not needed. They detach these blocks to obtain the smallest collection con-taining both buildings. (See fig. 14, method 2.) Some students may notice that these duplicate blocks represent the GCF of the two numbers.

Relating the lcM and the gcFIn my experience, most students tend to remember this method the best and use it most often in the long term, possibly because it can be interpreted as a simple computation. The details of the computation will depend on the order in which stu-dents manipulate the blocks. Detach-ing the GCF from one building and then attaching the resulting collec-tion of blocks to the other building is equal to dividing one number by the GCF and multiplying the result by the other number. In algebraic language, we may write this as

LCM(a, b) = a ÷ GCF(a, b) × b.

Attaching the buildings before de-taching the blocks that they have in common gives us

LCM(a, b) = a × b ÷ GCF(a, b).

Students may like to see some simple examples. For instance, since the GCF

of 24 and 18 is 6, the LCM may be calculated as

24 ÷ 6 × 18 = 4 × 18 = 72

or

18 × 24 ÷ 6 = 432 ÷ 6 = 72.

You may have noticed that the least common multiple of two numbers can sometimes be found by multiplying them. When can this be done? I encourage students to look closely at the formu-las to help them answer this question. When GCF(a, b) = 1, the formulas

become LCM(a, b) = ab; the numbers have no common factor other than 1, and their buildings have no blocks in common. Such pairs of numbers are called relatively prime.

Did any groups focus on one color at a time and decide how many of each color are needed to form the least common multiple? Students rarely discover this strategy, but I usually guide them through it.

If the first number contains four white blocks and the second number contains six white blocks, how many

Method 1

3711 13

2 2 2 2 23 37

1848 3276

Begin with either building

3711

2 2 2

1848

313

Attach just the blocks needed so that the block diagram for the other number (3276) is also contained

3711

2 2 272072

3

13The result is the leastcommonmultiple

Method 2Begin with either building

2 237

Detach the GCF of the two numbers

Attach the remaining blocks to the other building

3711

2 2 272072

3

13The result is the leastcommonmultiple2

11

3711

2 2 2

13

2 23 37

3276

Method 3 Select one color (factor) at a time. Choose the larger of the two collections of blocks.

white

red

yellow

green

blue

from 1848 from 3276 the larger collection

1848

2 2 2 2 2 2 2 2

3 3 3 3 3

7 7 7

11 none 11

none 13 13

attach to form the LCM

3711

2 2 272072

3

13

Fig. 14 many strategies evolve for building the lCm of two numbers, if their block diagram is known. this example demonstrates finding the lCm of 1848 and 3276.


3

57

2 2 2 237

233 3

5

1008

9450

GCF(a,b)

LCM(a,b)

2 2 2 2

2

33 3

33

55

7

7

75600

126

a

b

Attach a and b Attach GCF(a,b) and LCM(a,b)

2 2 2 233 355

7 7

233

2 2 2 233 355

7

233

7ab GCF(a,b) x LCM(a,b)=

006,525,9006,525,9

Fig. 15 Using the prime number blocks, rules emerge for finding the gCF and lCm. Specifically, for each color, the smaller number of blocks from a and b goes to the gCF and the larger number of blocks goes to the lCm, as in the examples shown for 1008 and 9450.

white blocks will their LCM contain? After some thought, students will see that it must contain six white blocks. All six are needed to ensure that the second number is contained in the LCM, but the additional four blocks from the first number are not neces-sary since they are already included. In general, for each color (prime factor), the LCM will always contain the larger of the two collections of blocks. For prime factorizations writ-ten in exponential form, this means that, for each base, we choose the larger of the two exponents. (See fig. 14, method 3.)

We can take this idea one step further. Using a similar line of ques-

tioning, students discover that the collection of blocks that the build-ings have in common (the GCF) is determined by the smaller of the two collections for each factor. Therefore, if we have block diagrams for two numbers, a and b, we may simultane-ously form the GCF and the LCM. We proceed, one color at a time, by attaching the larger collection to the LCM and the smaller collection to the GCF. Since this process uses each block in the two buildings ex-actly once, we can see that attaching the GCF and the LCM will produce the same result as attaching the two original numbers. (See fig. 15.) The formula is

GCF(a, b) × LCM(a, b) = ab.

This formula is equal to those found from the second strategy.

sUMMaRYConcrete representations of mathe-matical structures can engage stu-dents’ interest and help them visualize and internalize challenging concepts. By using building blocks to represent prime factorizations, my students have gained a deeper appreciation for the structure of the natural numbers. Along the way, they have discovered many interesting patterns, relation-ships, and procedures.

Some readers may be inter-ested in thinking about how the building-blocks model might be extended to other concepts, such as negative exponents and reciprocals, fractional exponents and radicals, and logarithms. Blocks may also be used to help visualize classic proofs such as the irrationality of 12 and the infinity of prime numbers. Ideas and related problems, activi-ties, and games are found at http://themathroom.org.

BiBliogRaPhYRobbins, Christina, and Thomasenia Lott

Adams. “Get ‘Primed’ to the Basic Building Blocks of Numbers.” Math-ematics Teaching in the Middle School 13 (September 2007): 122−27.

Zazkis, Rina, and Peter Liljedahl. “Understanding Primes: The Role of Representation.” Journal for Research in Mathematics Education 35 (May 2004): 164−86.

Jerry Burkhart, [email protected], teaches sixth-grade mathematics in the mankato area Pub-lic Schools in minnesota.

he is interested in finding new ways to use mathematical models to help students make sense of challenging concepts.

building primes

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