PoW-TER Problem Packet
Divisible by a Dozen
1. The Problem: Divisible by a Dozen [Problem #2960]
Take any three consecutive integers. Multiply the first times the square of the second, then
multiply by the third.
Example:
If you use 2, 3, 4 then you would have 2 * 32 * 4, which equals 72.
Try this with four different groups of three consecutive integers.
Is each result divisible by 12?
Use prime factorization to support your conclusions.
Extra: Show that this is always true no matter what three integers you start with.
Note: When typing your answer, you can use the character ^ to indicate that the next item is an exponent:
5*5 (5 squared) would be 5^2.
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2. About the Problem
Divisible by a Dozen is a problem #2960 in the Problems of the Week Library. It is a
challenging problem to be used with younger students who have learned some number
theory on prime factoring and pre-algebra or algebra students who can express the given
rule using variables. The problem states a rule and asks the students to try the rule on four
different sets of numbers. The student is to use prime factorization to support conclusions.
This problem encourages students to look at division another way. Discussions could
begin with the wealth of vocabulary: consecutive, integers, squaring a number,
divisibility, prime factors and factor trees. Prime, composite numbers, multiples, even, and
odd are also vocabulary that can be used in the introduction of the problem. Encourage
students to use the vocabulary in their written explanation.
I find this problem valuable and challenging to use with my sixth graders who study a unit
on variables and general patterns followed by a unit on rational numbers. The main goal
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for all students is to understand divisibility and support the conjecture of divisibility in
terms of prime factorization. A second goal for my sixth graders is to use variables to
describe a general pattern and to demonstrate the general pattern with four specific
examples. This unit on variable expressions is followed by rational numbers that will
naturally lead to reducing rational numbers by canceling common factors. Divisibility
could be seen as canceling all the prime factors of 12. My sixth graders struggle with the
extra section of this problem but it is an excellent way to get them to represent
consecutive integers with single variable, make connections with number theory and
communicate using mathematical vocabulary.
In pre-algebra, I would use this problem during our unit of study on rational numbers.
With these students my goal would be to review the writing of a rule in terms of variables
and to focus on rules of divisibility. With pre-algebra or algebra students I would require
the Extra. I would be looking for a discussion on the number theory that would ensure
division by 12. A second goal would be to encourage the students to write the rule in
terms of one variable expression using exponents and evaluate the expression correctly
using order of operations. In the specific examples with pre-algebra, I would expect them
to use both positive and negative values to evaluate the expressions. My goal for my
seventh and eighth graders would be the same as with my sixth graders to continue to
develop the concept that divisibility is the cancellation of all the prime factors of the
divisor and maybe see this concept in terms of variable expressions and number theory
about multiples.
With my algebra students this problem could be used in terms of defining a function using
one variable and focusing on the reasoning and proof and making connections with the
number theory they know. I would like to see them square a binomial correctly and
multiply the expression. Then they might hit a roadblock since in Algebra 1 students
usually do not have the skills to work with fourth degree equations. They should be able to
support the conjecture using number theory about multiples and factors. Students are
capable of reasoning about this problem. The question could then be presented to find
different ways to represent the different possible arrangements of the three consecutive
integers. Would a different representation of three integers make it easier to insure
divisibility by 12?
This is a problem built for differentiation and ripe for discussion at various levels.
The problem will help the teacher assess a student’s understanding of number theory.
Align this problem with the NCTM Curriculum Standards of Numbers and Operations
(Grade 6-8). Do the students understand numbers, understand patterns, understand ways
of representing numbers and understand the meaning of operations? Align this problem to
the NCTM standards of problem solving, reasoning and proof, and communication. Can
the student build new knowledge working on this problem? Can he or she develop and
support a mathematical argument? Can the student communicate to peers and teacher his
arguments and can they understand and question another’s argument?
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Divisible by a Dozen will give students the opportunity to build on mathematical concepts
if working alone or in a collaborative group? Is the product using the given rule always
divisible by 12? If it is divisible by 12 in the four specific examples given by the student,
can the student logically support the fact that the product will always be divisible by 12?
In order to solve the extra portion in the problem the student must call on his or her
knowledge of number theory, the meaning of divisibility and the various ways to represent
the consecutive numbers. At each level the teacher can help the students build new
mathematical knowledge through their grappling with this problem.
3. Common Misconceptions
Directions Misunderstood- Pattern Missed
Most students understood the meaning of consecutive numbers, but the majority used only
natural numbers and not integers. Many students were able to answer “yes” to the question
that the product from the given rule using consecutive integers is divisible by 12. Several
did not follow directions by giving four specific examples of the rule. The majority of
students missed the primary goal of the problem: the use of prime factorization to support
divisibility. For many the only concept of divisibility developed is that the quotient is a
whole number when dividing by 12.
No prime factorization – Pattern Missed.
The first group that I will do is 6*7²*8. It equals 2,352. Then we will divide it by 12. So
then 2,352/12 equals 96.
The second group I will do is 9*10²*11. 9*10²*11 equals 9,900. Then we have to divide
9,900 by 12. So then 9,900/12 equals 825. So again it works.
The third group of numbers I will 16*17²*18. So then 16*17²*18 equals 83,232. Now we
have to divide 83,232 by 12. So then 83,232/12 equals 6,936. So again it works.
The fourth group of numbers I'll do is 25*26²*27. That equals 456,300. Then we have to
divide 456,300 by 12. So then 456,300/12 equals 38,025. So it works.
My theory on how this works is that if you square the second consecutive number it will
always work because 1*2 (with the two representing the squared number. So then when
you try a prime factorization it will always work.
The student work above shows that the student knew the meaning of consecutive counting
numbers but not necessarily consecutive integers. The student was able to replicate the
example with four specific examples. The student missed the primary goal of the question
since the directions about supporting conclusions by prime factoring were not even
considered. Most likely the use of a calculator here is how the student knew divisibility.
The student may not yet be thinking abstractly. Although this student knew prime
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factorization was expected to support the explanation, there is no clue that the student
knows what to do with that information.
When giving this problem in my sixth grade, I block out the example to see if they can
translate the written rule to a specific examples. This student was successful. Leading
questions for this student would be: I wonder what are your prime factors. How can you
prime factor the products given in your examples presented? How can you prime factor
12? How can you prime factor your quotient? Is there a difference in the list of prime
factors for the product and the quotient? Is there a pattern? I would also encourage this
student to write a general pattern for the four specific examples given.
Factoring is not prime factorization - directions for extra also include consecutive
integers.
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This attempt at the extra demonstrates that the student misread the directions for showing
that the pattern worked for any set of three consecutive numbers. The original four
specific examples used consecutive whole numbers and were written out correctly. Here
the student lets you know that order of operations is performed. The student thinks
factored form is prime factorization. In the extra an example of the product divisible and
not divisible by 12 is given since consecutive numbers were not used. Having missed the
concept of using prime factorization to support divisibility, I would have paired this
student with the student above and ask them to prime factor the given product not
divisible by 12, prime factor 12 and see if they could make a conjecture as to when any
product is or is not divisible by 12. I wonder if that would help both students support
divisibility by 12 using prime factorization. I wonder if this might help them grapple with
the extra problem. I would wonder with this second student what squaring means and
what is a square unit. I would make sure that this concept is explored and corrected.
In both of the above solutions the students missed the primary goal of the problem to see
divisibility by 12 in terms of prime factorization.
Verbal model and specific example misunderstood.
Here is another example of a lack of understanding the directions for the four specific
examples to follow the given rule. Both the verbal model and the computational model
were misunderstood.
The student could not translate the initial problem correctly. The use of consecutive
numbers was accurate but it appears that instead of squaring the second number, the
second number was raised to a power, the power being the same number as the first factor.
Even though the original expression was not translated correctly the computation is
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correct. Each of these products is divisible by 12. This student understands divisibility by
12 implies 12 is a factor. This student did not show an understanding of prime
factorization although he understood divisibility. I would wonder if after the presentation
the student might be able to explain why all his expressions would also be divisible by 12.
Computation Problems – Vocabulary Errors
Generally most students knew order of operations and squared the second number first.
There might be a few who did not use consecutive integers or did not compute the product
correctly. The meaning of squaring is crucial to this problem.
Squaring is the problem
One needs to find out if the first student above knows the meaning of squaring and the
order of operations. I only presented you with one of the groups but all four examples by
this student reveal that the student doubled the second number (if order of operations was
followed) or doubled the product of the first and the second number. The second student
seems to know prime factors of 144 and 24 but again had trouble with squaring. It appears
that this student doubled the square of the second number. I would wonder if these
students know the squares of 6, 4, 3 and 2. I would ask them to write the squares of 6, 4,
3, and 2 in factored form.
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Squaring an integer - What is an integer?
I tried
4,5,6
7,8,9
-3,-2,-1
My work was
4*5^2*6
7*8^2*9
-3*-2^2*-1
I got
600
4032
-12
They were all divisible by 12.
When you try to do it with numbers that are not integers you get numbers that.....
12*13^2*14=28,392
it is 12 13 13 14 over 12
you cross out the 12s and you are left with 13 13 14 whose factor trees are 13*1, 13*1,
7*2 with the numbers being 13, 13, 7, 2, 1,1 which equals 37
At first I was excited to see an example of consecutive negative integers, but realized
further that this student made some vocabulary and computational errors whether using a
calculator or not. It appears the examples of integers used included positive and negative
values of single digit numbers. I would wonder what set of numbers 12, 13, and 14
belonged to. The student knew order of operations and appears to understand division as
the cancellation of common factors. The use of factor trees shows a process for getting
numbers down to primes but new learning can be built by wondering if the number 1 fits
the definition of a prime factor. I would also wonder what equals 37. If factoring, I
wonder why one would find a sum.
The answer to
is correct. This is not a correct translation of the written
rule above. The computation for the correct expression is:
. This is a common error made by students even if they use a
calculator and forget parenthesis. It is important that students know that
reads the
opposite of 2 squared whereas (
is negative 2 squared.
Understanding the directions to give four specific examples, consecutive integers,
squaring and prime factorization are essential to answering this problem and building a
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conceptual knowledge of divisibility. Supporting the conjecture through the use of prime
factorization is a primary goal of this PoW.
4. Sample Student Solutions/Discussion
The agenda is for the students to give four examples of the given rule, and I ask for a
general variable model from my youngest students. Communication is to confirm
divisibility by 12 and to support the conclusion of divisibility by using prime factoring.
Reasoning and proof using the general variable model can be seen in those who answered
the extra.
1. Four specific examples:
This student gave us four specific examples using positive consecutive numbers. The
computation is correct. The student gave us the product and then prime factored these
products. Divisibility is confirmed by showing us the prime factors of 12 the factors in
parenthesis exists in each of the products. I would wonder how the communication for
divisibility of 12 could have been expressed differently than showing prime factors of 2,
2, and 3. The next student gives us four specific examples and communicates how we
know the prime factorization will be divisible by 12.
Most students will not need support in following the four specific examples, once
presented with these specific problems. Those who did not use prime factorization to
support an argument would understand 12 as being 2*2*3. Understanding of prime
factorization as part of division will need a little support. Younger students would want to
do the division problem to be convinced of divisibility. Younger students understand
divisibility as having no remainder.
Support could be given by having them do the division and then find the product of the
factors above not in parenthesis. To support these students I would write the division
problem as a fraction in prime-factored form.
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2. Divisibility by 12 Defined
This student communicates immediately how one knows divisibility by 12. If the prime
factors 2, 2, and 3 are in the list of prime factors then the number is divisible by 12.
Reasoning to show divisibility by 12 no matter what three consecutive numbers are
chosen is explained in terms of the possible arrangements of even and odd numbers.
Squaring the only even number will guarantee 2* 2 in the list of prime factors. If odd is in
the middle the two even numbers in the consecutive list will guarantee 2*2 in list. The
student knows 3 is a factor in any three consecutive numbers. The exception to this
thinking is the three examples using 0 in the list of consecutive numbers. In these three
examples you will not find 2, 2, and 3 as factors. I would wonder if 0 is divisible by 12. I
would wonder what integer in the consecutive numbers listed in the exceptions is a
multiple of 3.
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Again in this presentation even the students who were not able to communicate why there
is divisibility by a dozen using the given rule could follow the thinking of every three
integers being a multiple of three. Even integers being multiples of two would be
understood by all. The presentation of the written response could be an “ah ha” moment
when students might think why didn’t I think of that. Only by continual exposure to
thinking like this will students improve in their reasoning abilities. To think of exceptions
to your rule is foreign to most students and they will need help in this regard.
3. Negative Consecutive Integers Work
As in the above example I have taken the liberty of clarifying the student copy that did not
format well in the following example. In the communication by this next student we get
an answer to the wondering if zero is a multiple of 12. Also in the four specific examples
this student uses two examples of negative consecutive integers. The prime factorization
work is done using the process of factor trees which is familiar to the middle school
student. The student does report the prime factorization using exponents and this student
define divisibility using the expression of 2^2 *3 being part of the prime factorization.
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This student defines divisibility by 12 as having the prime factors of 12 in the prime
factorization. The student also addresses one of the exceptions to his definition. The last
two prime factorizations are corrected. This is a happy mistake for we can lead the
students into the next example.
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The process of factor trees is familiar to most middle school students. Do all make the
connection to factor trees, prime factorization, and divisibility? Here again the
presentation might be the moment where the link is made.
The mistake in the process above is common especially when students advance to
multiplication and division of rational numbers. They want to multiple first and then
reduce. The next example could support the idea of cancellation when working with
fractions
4. Four sets – the product never found
Here the student relates divisibility by 12 to the multiples of 12. The student did not find
the product of the given rule. Just wrote the expression in factored form and prime
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factored each of the factors. Writing the expression as 12 times a number insures the
divisibility by 12. I would wonder how we can incorporate the exceptions noted above to
this format.
This progression of not multiplying first is key to understanding the process of
cancellation in a division problem and in working with fractions. Middle school students
need support here in order to see divisibility as the cancellation of common factors. Have
the students rewrite the examples above.
=
=
Divisibility is when there is no remainder. Divisibility is when you have multiples of a
number. Divisibility is canceling common factors.
5. Reasoning Statements to Build On –
Why Square the Middle Number
This student emphasis is on why you must square the middle number to insure there are
two factors of two. The idea of two cases is mentioned, having two even numbers in the
list or only one even in the list. In the second situation unless one squared the number
there might not be 2*2. I would wonder if there was only one even number would there
ever be a 2*2 in the middle number without the squaring.
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Properties of zero
The comment above shows the student reasoning about consecutive integers including
zero and negatives. If we have reasoned divisibility to be related to multiples of 12 we
know zero will be divisible by 12. This comment can lead to a discussion on the properties
of zero.
One must be careful with the above comment about negatives and wonder for the student
if the result will be negative and ask for a specific example, in order to revise this
statement.
The next student considers multiples of 3 and the different positions for even and odd
multiples of three.
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I would wonder how the exceptions of #2 could be worked in here and used to enhance
the student’s thinking.
The reasoning about this problem and the communication about mathematical ideas needs
much support in middle school. That is why it is important to present any attempt at
reasoning and lead the students to revise or clarify their thinking. Younger students need
encouragement in thinking about different possibilities or cases of the given problem.
What is different about this arrangement? Does it fit my original conjecture? How can I
revise my conjecture?
6. Use of variables
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Excitement builds in 6th or 7th graders if they can find another variable expression for
consecutive numbers, even if they get stuck after that. Each member of the class can pick
a different “a” and write down a, a+1 and a+2. I would wonder if everyone has three
consecutive numbers. Follow the rule. Is your number divisible by 12? I wonder what
would happen if everyone picked a different “n” would the above expressions give
consecutive integers. I wonder would the exceptions discussed in example #2 also work. I
wonder what would happen if you let a = the middle number. How would you express the
number before and the number after “a”? I wonder which expressions above could help us
recognize a factor of 2 squared. I wonder which expression above could help us recognize
a factor of 3.
Although finding a finite number of examples is not a proof, it is a start with those
beginning to develop abstract ability. Algebra students might be able to take these
different expressions of consecutive integers and use their multiplication of binomials
skills to prove 2, 2, and 3 are factors of the given rule.
5. Presentation Order
The presenting order above is deliberate to help students build on information gained as
we progress through the six topics. The first goal is achieved when example #1 gives four
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specific examples of the given rule and used prime factorization to show that 12 in prime
factored form was in the four examples.
The progression continues in example #2 when the student begins with the
communication of divisibility, gives four examples and then proceeds to explain why this
would work for all integers. The thinking applies number theory of even and odd and
multiples of three. Reasoning skills are developed to recognize the situation where the
product of three consecutive numbers is zero and is noted as an exception to the rule.
Student three continues the progression of the problem by using negative integers and
explaining the exceptions presented by the product zero. Student three uses factor trees for
finding prime factorization of the products. This leads to several computational mistakes.
This extra work in finding the product can be bypassed as seen in the next example.
Student four eliminates the need to find a product and simplifies the process by factoring
the factors. This student expands the concept of divisible by introducing the relationship
of multiples of 12. Using the vocabulary of multiples of 12 will allow the exceptions of
student #2 to fit the general pattern.
So having progressed from four specific examples of counting numbers our thinking has
expanded to use all integers in the specific examples. Seeing the factors of twelve in each
product implies divisibility since we can express the product as a multiple of 12.
Excerpts of student reasoning would be presented next. Students need support in this area.
So by presenting any and all excerpts from the students who attempted to support their
ideas will build on the collective knowledge of the class. Students need to apply theory,
develop and support an argument and refine the use of math vocabulary. The reasoning
power of defining a concept, applying properties of numbers, exploring different
combinations of consecutive numbers will enrich the conversation and open the eyes of all
to explore further.
The last presentation will be to present all the excerpts from students who made a general
pattern for the given function. One goal I have for my youngest to the oldest is to express
the specific and verbal models as a general pattern. Although a formal proof is beyond
most of my students, the reasoning and the arguments are not, if given support.
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6. Extension Lesson: Putting it all together
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This student, with some Algebra 1 skills, was able to do the first part of the problem.
Shown are four specific examples with the prime factorization of the product written as a
multiple of 12 and, therefore, divisible by 12. I would wonder and encourage the student
to compose a written explanation to explain this concept of divisibility.
The Algebra 1 skills are seen in the variable expressions, the squaring of the binomial, and
the multiplication of the polynomials. An attempt was made to express the polynomial in
another form, possibly to find a factor of 2*2*3. As seen in our other variable expressions
of three consecutive numbers this student might be able to revise and find the factors need
to show divisibility by 12.
What is lacking in this example is communication of ideas. What is valuable about this
attempt at the extra is the thinking behind evaluating this function when n = 1, 2, 3. We
need to understand the polynomial function is the product of the given rule and therefore
the range when using integers as the domain is multiples of 12. In an Algebra 1 class it
could lead to multiple representations by using a table from the graphing calculator or a
spreadsheet.
The lesson could extend by putting the function in y1 and setting the table to start with 0
and increase by 1. Looking at the table we would see consecutive integers and the range
consisting of integers. Most of us would have a hard time seeing the range as multiples of
12. I would wonder how we would know the number was divisible by 12. We could let
y2 = y1/(2*2*3) or y1/12.
From our discussion on the PoW we knew divisibility by 12 could be expressed as
multiples of 12. So let y3 = y2 * 12. Although this is not a formal proof we are building on
the knowledge gained and looking at multiple representations.
This form of representation as an extension to the problem can build on concepts of the
domain and range, and expressing relationships in terms of variables or functions. (see
table below)
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7. Extensions
Having worked on this problem and building a presentation on students’ work similar to
the one above, I could extend the lesson for the 6th and 7th grade students to support a
different pattern. Can the students build on their learning from the PoW Divisible by a
Dozen to support the argument below?
A Different Pattern
Even though this is not the pattern we were looking, encourage pattern discovery and ask
the class to see if this holds true with the specific examples each child used. Pose the
question as to why this would always work with three consecutive integers. This would
be a great pattern for Algebra 1 students and Pre-Algebra students who can multiply
binomials to try and show algebraically why it would work with all sets of three
consecutive integers as an extension to our answering the extra problem. Especially after
our discussion on different ways to express three consecutive integers students might be
able to reason that (n-1) * n2* (n+1) would verify the pattern suggested above.
8. Conclusion
Hopefully after having worked through this problem students will have developed a sense
of divisibility as related to sharing common factors and cemented their understanding of
integers. Students would begin to understand developing and supporting a mathematical
idea through specific examples and then awaken the need for a general format.
Communicating mathematically in written and discussion formats while working on this
problem with peers and teachers will increase vocabulary development and the collective
enrichment of all. Enjoy Divisible by a Dozen.
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