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.
Building
Numbers
fromPr mes
Dex Image/Photolibrary
Jerry Burkhart
p
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.
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.
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 elements 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 relationship to other numbers.
For the last few years, I have used the activities described here over a twoweek 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 centimeter 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
Vol. 15, No. 3, October 2009
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Mathematics Teaching in the Middle School
157
1
Fig. 1 A listing of prime numbers upFigure
to 20 and their representative color block
Color
Prime Number
white
2
3
5
10
yellow
7
5
green
11
3
blue
13
3
purple
17
2
brown
19
2
red
orange
Suggested Quantity
per Group
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
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.
5
5
black*
*Black blocks may be used to
represent any prime number
greater than 19.
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 progresses and the students discover the
types of building blocks needed, each
color will be assigned an appropriate number so that we can build the
counting numbers. (See fig. 1.)
This article will use a questionand-answer format to mirror the content 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.
158
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 factors 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
Mathematics Teaching in the Middle School
●
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.
Vol. 15, No. 3, October 2009
As groups understand the process,
they begin to work more independently. 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. Attaching three white 2 blocks as 2 × 2 × 2
represents 8.
Students continue building the remaining 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? Students 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
very important idea: Counting numbers 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 counting 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 concern 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-ata-time strategy, illustrated in figure 3.
Others realize that they are finding
prime factorizations and building factor trees. (If so, I introduce the necessary vocabulary.) Still others write the
number they are building as a product
of two factors, look at their previously built numbers, and attach copies
Fig. 2 Building the counting numbers 2 through 12. The colored blocks help to visualize the factors of composite numbers and identify prime numbers.
Natural Number
Block Representation
(Prime Factorization)
2
2
3
3
4
2 2
5
5
6
3
2
7
7
8
22 2
9
3 3
10
5
2
11
11
12
3
2 2
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.
Vol. 15, No. 3, October 2009
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Mathematics Teaching in the Middle School
159
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.
1
2
3
5
6
7
8
9
10
7
2 2 2
3 3
5
2
2
3
2 2
5
3
2
21
31
22
51
21 x 31
71
23
32
21 x 51
11
12
13
14
15
16
17
18
19
20
11
3
2 2
13
7
2
5
3
2 2 2 2
17
3 3
2
19
5
2 2
21 x 71
31 x 51
24
171
21 x 32
191
22 x 51
24
25
26
27
28
29
30
5 5
13
2
3 3 3
7
2 2
29
111
22 x 31
131
21
22
23
7
3
11
2
23
3
2 2 2
31 x 71
21 x 111
231
23 x 31
52
21 x 131
33
22 x 71
291
31
32
33
34
35
36
37
38
39
40
31
2 2 2 2 2
11
3
17
2
7
5
3 3
2 2
37
19
2
13
3
5
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
43
11
2 2
5
3 3
23
2
47
3
2 2 2 2
7 7
5 5
2
21 x 231
471
24 x 31
72
21 x 52
41
411
7
3
2
1
x
2 31 x 71
431
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 attach the two white 2 blocks to make 4
and the two red 3 blocks to make 9.
Why do you think that prime numbers 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 defined 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 activities 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
160
4
22 x 111
32 x 51
could use as many or as few 1 blocks
as we like to build any counting number. 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 STRUCTURE
Students 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, numbered 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-
Mathematics Teaching in the Middle School
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Vol. 15, No. 3, October 2009
5
3
2
21 x 31 x
51
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 below 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
appearances seem to be distributed
almost randomly throughout the grid.
Can you see a way to predict the location 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, mathematicians have never found a simple way
to predict the next prime.
Certain types of patterns can be
found. For example, do you notice anything 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 buildings 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
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.
11
11
2 2 2
23 x 111
88
attach
5 5
3
x
31 x 52
x
75
5 5
3
2 2 2
=
=
23 x 31 x 52 x 111
6600
5 5
3
11
2 2 2
11
5
2
10
x
10
5 5
3
2 2 2
11
5
2
3
2
x
66
=
6600
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:
a grid posted in the classroom. It can
be a wonderful resource for identifying prime numbers, verifying multiplication 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.
1. How can we see that every multiple 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
MULTIPLICATION, DIVISION,
AND EXPONENTS
The next step asks students to analyze
the blocks for multiplication.
Vol. 15, No. 3, October 2009
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How can we use blocks to show the
multiplication of composite numbers?
Attaching individual blocks (prime
numbers) means that the two numbers 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.
Mathematics Teaching in the Middle School
161
Fig. 6 Dividing 60 by 10 occurs by detaching blocks.
5
3
2 2
detach
(22 x 31 x 51)
60
5
2
3
2
/
(21 x 51)
/
10
21 x 31
=
6
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.
2 22
attach
23
x
2 2
2 2 2 2 2
22
=
25
Fig. 8 Detaching blocks and seeing the exponent decrease helps students realize that
30 = 1 when there are no blocks left.
3
27
=
33
(3 “3” blocks)
3 3 3
(detach 1 “3” block)
(divide by 3)
9
=
32
(2 “3” blocks)
3
3 3
(detach 1 “3” block)
(divide by 3)
3
=
31
3
(1 “3” block)
(detach 1 “3” block)
(divide by 3)
1
=
3
30
(0 “3” blocks)
If we multiply numbers by attaching blocks, how would we divide them?
Students usually suggest that blocks
should be separated or detached.
Avoid phrases that suggest subtraction, 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 exponential 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:
am
= a m−n
an
No Blocks
Susie : 2is in last
parasome
beforestudents
bibliogrraphy
Since
may be confused
by the addition and subtraction in our
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Mathematics Teaching in the Middle School
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Vol. 15, No. 3, October 2009
2
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 exponential 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 contradicted by figure 8. The number 27
is 33, so we build it with 3 red blocks.
Each time we divide by 3, the exponent 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 attaching blocks to no blocks would represent 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 Arithmetic 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
Fig. 9 The factors of 210 can be found by detaching a single block or clusters of
blocks.
7
5
3
2
7
5
3
2
7
5
3
2
7
5
3
2
7
5
3
2
5
3
7
2
105
x
2
7
x
30
7
5
3
2
7
5
2
3
7
5
3
2
7
5
3
2
7
5
3
2
15
x
14
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 MULTIPLES
Building 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
Vol. 15, No. 3, October 2009
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7
5
3
2
no blocks
70
x
3
35
x
6
7
5
3
2
7
3
2
5
7
5
3
2
7
3
5
2
42
x
5
21
x
10
210
x
1
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 factor 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, guiding 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 represent 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
Mathematics Teaching in the Middle School
163
Fig. 10 Multiples of a number are formed by attaching blocks to its diagram. In this
case, multiples of 12 are found.
3
2 2
2
3
2 2
3
3
2 2
2 2
3
2 2
5
3
2 2
3
2 2
12 x 1 = 12
3
2 22
12 x 2 = 24
(attach a “2” block)
3 3
2 2
12 x 3 = 36
(attach a “3” block)
3
2 22 2
5
3
2 2
(attach no blocks)
12 x 4 = 48
12 x 5 = 60
(attach 2 “2” blocks)
(attach a “5” block)
Fig. 11 Finding the greatest common factor can be found easily when defined as “the
largest collection
of blocks
is contained
in both
blockofdiagrams”
for each number.
Building
the that
Greatest
Common
Factor
36 and 54
3 3
2 2
3 3 3
2
36
54
Common factors of 36 and 54 are collections of blocks
that are contained in both of the above block diagrams:
the “empty”
collection of
blocks
1
2
2
3
3
3
2
6
3 3
9
3 3
2
18
The above collection containing the most blocks is
the Greatest Common Factor of 36 and 54:
3 3
2
164
18
Mathematics Teaching in the Middle School
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Vol. 15, No. 3, October 2009
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 seeing 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 attaching 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 discovering and describing the mathematical relationships inherent in greatest
common factors (GCF) and least
common multiples (LCM).
Since factors are formed as subcollections 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 common factor is found to be the unique
such collection containing the most
blocks. In short, the GCF is represented as “the largest collection of
blocks that is contained in both block
diagrams.” (See fig. 11.)
THE LCM
Since we form multiples by attaching
blocks to the original building, every
multiple of a number will contain
the original building. Thus, a common 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.) Removing 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 multiples 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 following 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 contained in the second number but “missing” from the first. (See fig. 14, method
1.) In my experience, this is the most
common strategy that students adopt.
When students imagine how difficult this computation would have been
had they made long lists of multiples
and looked for common numbers, they
appreciate the advantages of
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.
3
2
5
3
6
15
Common multiples of 6 and 15 must contain
both of the above block diagrams.
For example:
5
3 3
2
5 5
3
2 2
5
3
2
7
5
3
2
90
300
30
210
The collection of this type containing the fewest possible blocks is
the Least Common Multiple of 6 and 15.
5
3
2
30
If any block is removed from this building, it will no longer be a common multiple.
Fig. 13 Using blocks to represent the greatest common factor and the least common
multiple of numbers M and N
Factor of N
Multiple of N
A collection of blocks
contained in the block diagram
of N
A collection of blocks
containing the block diagram of
N
Common Factor of M and N
Common Multiple of M and N
A collection of blocks
contained in the block
diagrams both of M and N
A collection of blocks
containing the block diagrams
of both M and N
Greatest Common Factor
of M and N
The largest collection of blocks
contained in the block
diagrams of both M and N
Least Common Multiple
of M and N
The smallest collection of
blocks containing the block
diagrams of both M and N
Vol. 15, No. 3, October 2009
●
Mathematics Teaching in the Middle School
165
using prime factorizations. This type
of strategy is often suggested to algebra
students who are trying to find least
common multiples of the denominators of algebraic fractions. We cannot
generally make ordered lists of multiples of algebraic expressions.
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.
11
13
7
3
2 2 2
7
3 3
2 2
1848
Did anyone attach the original
numbers, then eliminate extra blocks?
Some groups think of attaching both
original buildings, since this will ensure 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 containing 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 GCF
In 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 students manipulate the blocks. Detaching the GCF from one building and
then attaching the resulting collection 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 detaching 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
166
3276
Method 1
13
Begin with either building
3
11
7
3
2 2 2
Attach just the blocks needed
so that the block diagram for
the other number (3276) is
also contained
The result
is the
least
common
multiple
Attach the remaining blocks to the
other building
Begin with either building
7
3
2 2 2
1848
Method 3
7
3 3
2 2 2
72072
1848
Method 2
11
13
11
7
3
2 2
13
11
7
3 3
2 2
2
Detach the
GCF of the
two numbers
The result
is the
least
common
multiple
13
11
7
3 3
2 2 2
72072
3276
Select one color (factor) at a time. Choose the larger of the two collections of blocks.
from 1848
from 3276
the larger collection
attach to form the LCM
blue
none
13
13
green
11
none
11
yellow
7
3
2 2 2
7
3 3
2 2
7
3 3
7
3 3
2 2 2
2 2 2
72072
red
white
of 24 and 18 is 6, the LCM may be
calculated as
24 ÷ 6 × 18 = 4 × 18 = 72
13
11
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.
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 formulas to help them answer this question.
When GCF(a, b) = 1, the formulas
Mathematics Teaching in the Middle School
●
Vol. 15, No. 3, October 2009
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
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.
a
GCF(a,b)
7
33
2 2 2 2
7
33
2
1008
126
LCM(a,b)
b
7
5 5
333
2
7
5 5
333
2 2 2 2
9450
75600
Attach GCF(a,b)
and LCM(a,b)
Attach a and b
ab
7
5
3
2
7
5
33 33
2 2 2 2
=
9,525,600
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 necessary 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 written 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-
GCF(a,b) x LCM(a,b)
7
5
3
2
7
5
33 33
2 2 22
9,525,600
This formula is equal to those found
from the second strategy.
SUMMARY
Concrete representations of mathematical structures can engage students’ 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, relationships, and procedures.
Some readers may be interested 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, activities, and games are found at
http://themathroom.org.
BIBLIOGRAPHY
tioning, students discover that the
collection of blocks that the buildings 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 simultaneously 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 exactly 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
Vol. 15, No. 3, October 2009
GCF(a, b) × LCM(a, b) = ab.
●
Robbins, Christina, and Thomasenia Lott
Adams. “Get ‘Primed’ to the Basic
Building Blocks of Numbers.” Mathematics 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, jburkh1@
isd77.k12.mn.us, teaches
sixth-grade mathematics
in the Mankato Area Public Schools in Minnesota.
He is interested in finding new ways to
use mathematical models to help students
make sense of challenging concepts.
Mathematics Teaching in the Middle School
167