JR Task Analysis
Sum of Consecutive Natural Numbers
Sum of Consecutive Natural Numbers Task
Find all the numbers less than 49 that can be written as the sum of two or more
consecutive natural numbers. Find, if any, the numbers greater than zero that
cannot be written as the sum of consecutive natural numbers.
CCSSM
Standards for
Mathematical Practice:
1, 2, 3, 6, 7, 8
Content Standards:
•
•
Operations and
Algebraic Thinking
Statistics and
Probability
Some numbers, such as 21 can be written as the sum of 2
consecutive natural numbers:
21 = 10 + 11.
Some numbers can be written as the sum of consecutive
natural numbers in more than one way:
39 = 19 + 20 39 = 12 + 13 + 14 39 = 4 + 5 + 6 + 7 + 8 + 9.
Some numbers cannot be written as the sum of 2 consecutive natural numbers.
Determine which numbers less than 50 can be written as the sum of consecutive natural
numbers. As you make this determination, keep a record of all your work. We will need
to refer to that as we discuss the problem. Also think about:
Which of these can be written as such sums in more than one way?
Which cannot be written as the sum of consecutive natural numbers?
After exploring the possibilities, make some conjectures related to this task.
Make some statements about the methods you used to find the possible sums.
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JR Task Analysis
Sum of Consecutive Natural Numbers
The Mathematics of the Task
The Consecutive Natural Numbers Task (CNNT) is an excellent application of inductive
reasoning, where one goes from specific numerical examples to formulating a
generalized statement. To take advantage of the opportunities for the mathematical
practice standards offered by this task, teachers must be able to see beyond individual
cases and express patterns in general terms. The task can be used to develop concrete
reasoning with numbers and bridge to more abstract conceptions involving algebra and
using symbolic notations. It is this transition that makes the problem so rich. Formal
inductive proofs aren't necessary for achieving mathematical depth with this task, but
developing generalized statements is crucial to making the transition from concrete
examples to symbolic abstraction. These abstractions may be in the form of verbal
statements, at first, and then progress to algebraic expressions.
Attention to communication through notation is essential as is the introduction of
symbols in order to express generalized statements. Most importantly, definitions of
consecutive natural numbers, odd numbers, and multiples of a number form the basis of
the reasoning upon which the proofs are built.
The first generalization that usually arises is that the sum of two consecutive natural
numbers is an odd number. Looking at examples reveals this pattern and leads to a
preliminary proof of that statement. This builds on the structure of consecutive natural
numbers:
1 + 2 = 3 1 + (1 + 1) = (1 + 1) + 1 = 2(1) + 1 2 + 3 = 5 2 + (2 + 1) = (2 + 2) + 1 = 2(2) + 1 3 + 4 = 7 3 + (3 + 1) = (3 + 3) + 1 = 2(3) + 1 The preliminary proof might describe the relationship between two consecutive natural
numbers that could apply more generally:
"When you add a number to the next number, you are adding a number and that
number plus one, which is twice the original number, plus one."
Here, 3 is the original number and 4 is the original number plus one (4 = 3 + 1).
3 + 4 = 7
3 + (3 + 1) = 7
This kind of discussion motivates the usefulness of variables. Specifically, how might we
name a number and a number plus one in general terms, rather than using a specific
example like 3.
What if we call the original number 𝑛? How would we write the original number
plus 1?
𝑛 + 1 If we are to add these two numbers (now written in abstract terms using
variables) we find:
Original number + original number plus one = two of the original number plus 1 𝑛 + (𝑛 + 1) = 2𝑛 + 1 CNNT
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JR Task Analysis
Sum of Consecutive Natural Numbers
We can then proceed to adding three consecutive natural numbers, as described in the
CNNT, using what we have learned from adding two consecutive natural numbers as a
starting point – the introduction of a variable as described in the discussion of the sum
of 3 consecutive natural numbers.
Some may be able to extend this approach from two numbers to three numbers from
the symbolic notation:
original number + original number plus one + original number plus two 𝑛 + (𝑛 + 1) + (𝑛 + 2) Others may need to go back to the concrete example and use 3, 4 and 5:
If 3 is the original number, 4 is the original number plus one, and 5 is the original
number plus two, then we have
3 + 4 + 5 = 12 3 + (3 + 1) + (3 + 2) = 12 We then can replace (or substitute) 𝑛 for the original number (3) and get the following
expression:
𝑛 + (𝑛 + 1) + (𝑛 + 2) Common numbers or terms can be combined, so we combine all the 𝑛’s (there are three
of them) and all the whole numbers (1 and 2) to get:
3𝑛 + 3. Another route to solving the CNNT is by exploring multiple examples to notice the
pattern. To go from examining the sum of three consecutive natural numbers to the
generalization that such a sum is a multiple of 3 or three times the middle number
demonstrates the power of inductive reasoning. Looking at the sums of three
consecutive natural numbers we see pattern:
3 + 𝟒 + 5 = 12 = 3(𝟒) 7 + 𝟖 + 9 = 24 = 3(𝟖) 10 + 𝟏𝟏 + 12 = 33 = 3(𝟏𝟏) All three sums are multiples of 3 and that sum is 3 times the middle number. The
question is: Why is that statement true. Here the explanations begin with using the
numbers. It is our job to help our students begin to make a more abstract presentation.
Looking at the first sum, we can see that the addends are actually:
3 + (3 + 1) + ([3 + 1] + 1) This is the structure of consecutive natural numbers, that each number is one more than
the previous number.
Taking a 1 from the third addend and adding it to the first addend we get:
3 + 1 + 3 + 1 + 3 + 1 or three [3 + 1]. CNNT
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JR Task Analysis
Sum of Consecutive Natural Numbers
The question becomes, “Is this pattern true for every sum of three consecutive natural
numbers?”
Further exploration of numerical examples results in the same pattern: the sum of three
consecutive natural numbers is always 3 times the middle number.
[Note: With early elementary grades, you could stop here and then explore other
patterns.]
Using a series of questions can help build to an algebraic formulation.
•
•
•
•
•
•
Does it matter what number you start with? No.
If it does not matter what the first of the three is, how do mathematicians say that
they do not care what the number is? They use a variable.
How would you write the sum of three consecutive natural numbers using a
variable? 𝑛 + (𝑛 + 1) + 𝑛 + 1 + 1 or 𝑛 + (𝑛 + 1) + (𝑛 + 2).
If the second formulation is given, the question becomes: “From where did the 2
come?” This leads to the first expression of the sum.
How can we show that the sum is three times the middle number? We take a 1
from the third number and give it to the first number.
This gives us [𝑛 + 1] + [𝑛 + 1] + [𝑛 + 1] which can be summarized as 3[𝑛 + 1].
This careful formulation leads to an algebraic explanation of the generalization. It is built
on the structure of consecutive natural numbers. Notice we did not get into the use of
the associative and commutative properties of addition. Such discussions may distract
from the reasoning in the proof. Although naming properties of numbers are important,
here the focus is on the logic rather than the names of those properties (and they can
be named later). Instead, we just use the idea that when you have a series of addends
you can add them in any order.
The development of a generalization for the sum of any odd number of consecutive
natural numbers follows the formulation above. Things are more complex when we look
at the sums of an even number of consecutive natural numbers. Looking at sums of 4
consecutive natural numbers is often confusing if the sums are chosen arbitrarily
instead of being built up systematically.
3 + 4 + 5 + 6 = 18 7 + 8 + 9 + 10 = 34 Building up the sums in order does provide a hint to a generalization:
1 + 2 + 3 + 4 = 10 2 + 3 + 4 + 5 = 14 3 + 4 + 5 + 6 = 18 4 + 5 + 6 + 7 = 22 5 + 6 + 7 + 8 = 26 When listed in order, the pattern of an increase of 4 is more apparent.
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JR Task Analysis
Sum of Consecutive Natural Numbers
When we shift the first number by 1, the sum increases by 4. We can explore this
pattern using variables:
𝑚 + [𝑚 + 1] + [𝑚 + 2] + [𝑚 + 3] = 4𝑚 + 6 [𝑚 + 1] + [𝑚 + 2] + [𝑚 + 3] + [𝑚 + 4] = 4𝑚 + 10 What we have done here is subtract 𝑚 and added 𝑚 + 4. In essence the sum
increases by 4. This verifies that the sums do increase by 4.
Drawing a parallel to the sum of 3, 5, or 7 consecutive natural numbers, we know that
the sum of that series of consecutive natural numbers is 3, 5, or 7 times the middle
number. The question is: “Can we find a formula for the sum of 4 consecutive natural
numbers that is 4 times a number?" Looking at the sums we found:
1 + 2 + 3 + 4 = 10 = 4[2.5] 2 + 3 + 4 + 5 = 14 = 4[3.5] 3 + 4 + 5 + 6 = 18 = 4[4.5] 4 + 5 + 6 + 7 = 22 = 4[5.5] 5 + 6 + 7 + 8 = 26 = 4[6.5] Each sum is 4 times the mean or average or median of all four numbers. The question
now becomes: “Does a similar formula work for the sum of 6, 8, and 10 consecutive
natural numbers?” It does. So we can make specific generalizations as well as the
super generalization about the sum of an even number of consecutive natural numbers.
An algebraic explanation of this occurs requires greater facility with algebraic
expressions than it is probably worth the time.
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JR Task Analysis
Sum of Consecutive Natural Numbers
Solutions
The aim of working on the Consecutive Natural Numbers Task is not to come up with
full proofs, but rather to arrive at intermediate conjectures, construct viable arguments
for these conjectures, critique reasoning, and whet teachers' curiosity about
mathematics not yet done. Many conjectures arise from the numerical explorations. The
following are statements of some common conjectures.
Conjecture
Notes for proving the conjecture
The sum of two consecutive
natural numbers is an odd
number.
The consecutive natural numbers can be expressed
as 𝑚 and 𝑚 + 1. The sum is 𝑚 + (𝑚 + 1) or 2𝑚 + 1.
This is an even number plus 1, which is an odd
number.
The sum of three consecutive
natural numbers is a multiple
of 3. (Or: The sum of three
consecutive natural numbers is
divisible by 3.) The sum also is
three times the middle number.
The consecutive natural numbers can be expressed
as 𝑛, 𝑛 + 1, and (𝑛 + 1) + 1; their sum can then be
rearranged to 𝑛 + (𝑛 + 1) + [(𝑛 + 1) + 1] = 𝑛 +
1 + (𝑛 + 1) + (𝑛 + 1) = 3 (𝑛 + 1). Another
approach is to consider the middle number as n. The
three numbers are (𝑛 − 1), 𝑛, and (𝑛 + 1).
If the number of addends is
odd, the sum is the number of
addends times the "middle"
addends.
We can use the above approach to show this.
Another approach to this is to use the middle number
as the variable m and have the sequence:
𝑛 – 2, 𝑛 – 1, 𝑛, 𝑛 + 1, 𝑛 + 2, so 𝑛 – 𝑘 and 𝑛 + 𝑘 are
the first and last numbers where 𝑘 is the number of
numbers before or after the middle number. This
formulation takes advantage of additive inverses.
If the number of addends is
even, the sum is the number of
addends times the average of
the two "middle" addends.
As shown in the Mathematics of the Task section,
the numerical approach leads to this conclusion. The
analogy to this is finding the median of an even
number of data points. The algebra for this can be
complicated and we suggest only doing that with high
school students who are comfortable with algebraic
manipulations.
No powers of 2 can be written
as a sum of consecutive
natural numbers.
In fact, a natural number can be written as a sum of
consecutive natural numbers if and only if it is not a
power of 2. Proving this in full detail is not easy, and
in all but advanced high school mathematics or
college mathematics major classes would be beyond
the scope of the students. A proof is provided as a
downloadable resource on the web site.
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JR Task Analysis
Sum of Consecutive Natural Numbers
Discussion of Consecutive Natural Numbers Task: Where to
Begin
The first step in beginning work on this task is to make certain the teachers understand
the meaning of natural numbers and consecutive. A discussion of this is based on
fundamental definitions. Ask questions like: “How do the sets of whole numbers and the
natural numbers differ?” You might also want to include the set of integers in this
question. Consecutive only applies to the natural numbers and integers. They are
consecutive numbers that differ by 1.
A basic approach is to provide the teachers with a list of all the numbers from 1 to 49
and have them find all the sums of consecutive natural numbers that equal a given
number. We have provided a worksheet for organizing this work. Following this work,
the numbers need to be organized according to whether they are the sum of 2
consecutive natural numbers, 3 consecutive natural numbers, etc. It is important to have
the actual sums written. A more structured approach can be used by providing an
organized worksheet with column headings, “sum of two consecutive natural numbers,”
“sum of three consecutive natural numbers,” up to nine consecutive natural numbers,
where each natural number with its accompanying addends is placed in one or more
columns. This structured approach will take less time and eliminates the need to
reorganize the data, an important part of doing mathematical work. We do need a
complete list for the numbers 1 through 49 to answer the question which numbers
cannot be written as the sum of consecutive natural numbers. Whether you use an
open-ended or more structured approach, it is important that teachers keep a record of
what they have found. Teachers can work individually or in small groups on the task.
Generalizations should be written for the patterns that teachers find. These
generalizations can be prompted by asking teachers to summarize what they noticed.
Each generalization should be exemplified using numbers. Then they should be
expressed using numbers in a form such as 6, 6 + 1, 6 + 2, …. As a last step variables
should be used. The first generalization that often is noted is that the sum of 2
consecutive natural numbers is an odd number. The proof for this is presented in the
Mathematics of the Task section.
A key feature of the Consecutive Natural Numbers Task (CNNT) is the transition from
specific numerical examples to an algebraic formulation. For the sum of three
consecutive natural numbers, teachers may notice that these sums are all multiples of
3. Collecting a couple of examples supports this.
3 + 4 + 5 = 12 8 + 9 + 10 = 27 Asking, “Why do you think that is true?” will prompt various answers all of which can be
explored. Eventually, we may come to a statement like, “Take 1 from the third number
and add it to the first number. This gives you three of the middle number which is a
multiple of 3.” That is:
3 + 4 + 5 is equivalent to (3 + 1) + 4 + (5 – 1) which is equivalent to 4 + 4 + 4.
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JR Task Analysis
Sum of Consecutive Natural Numbers
This conclusion gives more information than the starting point that the sums are
multiples of 3. This conclusion can now be more precise: not only is the sum of three
consecutive natural numbers a multiple of 3, but in fact, “The sum of 3 consecutive
natural numbers is three times the middle number.”
Exploration of this statement can continue. First more numerical examples, preferably in
numerical order (e.g., 1 + 2 + 3, then 2 + 3 + 4, etc.), are given. Then the transition to
algebra can be made. The key question is: “What if we were to take any three
consecutive natural numbers, what would they look like?”
The usual response is:
𝑛, 𝑛 + 1, 𝑛 + 2.
The follow-up question is, “How did you get each of these numbers in the sequence?”
The answer may go something like, “I picked 𝑛 for the first number, I added 1 to 𝑛 to get
the second number, and added 1 to that number to get the third.”
Rewriting the symbolic expressions to emphasize what is actually said, we get:
𝑛, 𝑛 + 1, [(𝑛 + 1) + 1]. This statement needs to be reconciled with 𝑛, 𝑛 + 1, 𝑛 + 2 which is easily done.
We then can express the statement as a sum:
𝑛 + 𝑛 + 1 + 𝑛 + 1 + 1 ,
which can also be expressed as:
{𝑛 + 1} + {𝑛 + 1} + {[(𝑛 + 1) + 1] − 1} = {𝑛 + 1} + {𝑛 + 1} + {𝑛 + 1} = 3{𝑛 + 1} This expression 3{𝑛 + 1} is both a multiple of 3 and 3 times the median or “middle
number” in the set. It is a judgment call as to whether or not you have the teachers
justify each step by stating the repeated use of the Associative and Commutative
Properties for Addition. Such an emphasis may distract from highlight the reasoning
used.
For an alternate explanation, someone may suggest using the variable, n, for the middle
number. Then the consecutive "numbers" are {𝑛 – 1}, 𝑛, {𝑛 + 1} and their sum is:
{𝑛 – 1} + 𝑛 + {𝑛 + 1}
or
3𝑛.
This representation of the consecutive numbers has the advantage of containing fewer
manipulations than using 𝑛, 𝑛 + 1, and 𝑛 + 2. Examining sums of five consecutive
natural numbers, and seven consecutive natural numbers usually results in a statement
that for any sequence with an odd number of addends:
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JR Task Analysis
Sum of Consecutive Natural Numbers
“If the number of addends is odd, the sum of the consecutive natural numbers is
the number of addends times the middle number.”
This conjecture can be proven using analogous thinking and algebraic expressions from
the discussion of the sum of three consecutive natural numbers. This reasoning can be
expanded to look at the sum of any odd number of consecutive natural numbers.
The next question is: “What about the sum of an even number of consecutive natural
numbers?” The sum here is: “The average of the two middle numbers multiplied by the
number of consecutive natural numbers.” If teachers are familiar with statistics, a
reference to taking the mean/average of the two middle numbers is a step in finding the
median of an even number of numbers helps here (Note: The median of a set of
numbers is finding its middle number when the numbers are arranged in numerical
order. If there is an odd number of numbers in the set, the median is the middle number
of the list. If there is an even number of numbers, there is no middle number so we take
the mean/average of the two numbers in the middle).
Teachers will notice that the numbers that cannot be written as the sum of consecutive
natural numbers are: 1, 2, 4, 8, and 16. Teachers then can use process of elimination to
show that these five numbers cannot be written as the sum of consecutive natural
numbers. Some participants will recognize these as powers of two: 20, 21, 22, 23, 24 and
25 and conjecture that, "no power of 2 can be written as the sum of consecutive natural
numbers." That is as far as we will go in this analysis. This last conjecture can be
proven but it is beyond the scope of the task. That in and of itself is powerful. There are
some things about which we are absolutely convinced are true but yet may not have the
tools to prove. This statement remains a conjecture until it is proven. A proof is provided
in a separate document.
Using the Consecutive Natural Numbers Task with K – 12
Students
The Consecutive Natural Numbers Task can be used in grade 1 and on with different
goals regarding both the generalizations and the proofs of these generalizations. The
basic skill needed to begin thinking about the consecutive natural numbers task is to be
able to add natural numbers. First and second graders will notice that the sum of 2 ccn
is an odd number. Simply making that generalization is the first step in developing
students’ reasoning skills. Second graders can begin to explain why this is true, if they
have a more formal notion of even and odd numbers.
In grade three, once students know their multiplication facts and are more proficient in
adding a series of natural numbers, students can make the generalizations about the
sums of an odd number of consecutive natural numbers and notice that there is a string
of numbers for which no one can find a sum of consecutive natural numbers. They may
not be able to define the sequence 1, 2, 4, 8, 16, 32 as powers of two, but will be able to
describe how each number is twice, or the double of, the one before it. Most likely they
will state the generalization that the sum of 3, 5, or 7 natural numbers is a multiple of the
number of addends. Their proofs of these conjectures will be based on induction by
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JR Task Analysis
Sum of Consecutive Natural Numbers
looking at the sums of a given number of consecutive natural numbers. Some third
graders may notice that the sum of:
3, (3 + 1), and (3 + 2)
can be written as:
3+4+5
and can talk about sharing a 1 from the 5 or (3 + 2) with the 3. That makes the sum:
4+4+4
which reinforces the multiplication fact: 3 times 4. Students may articulate that the sum
is three times the middle number.
By Grade 4, using the questioning approach presented in the Mathematics of the Task
and the Discussion of the Consecutive Natural Number Task: Where to Begin
sections, students can begin to prove the statement in a more general way, using a
variable for one of the numbers. One way of extending their thinking is, after using a
class discussion to “prove” the generalization for the sum of 3 consecutive natural
numbers, is to ask the students to make similar proofs for the sums of 5 or 7
consecutive natural numbers. In later grades, students can tackle a generalization for
the sums of an even number of consecutive natural numbers.
If the task is first given in middle school or high school, the students should be able to
reach the formal proof stage quickly for all the generalizations discussed above. The
proof that there are no sums of consecutive natural numbers that equal a power of two
is very advanced and can be assigned to advanced high school students who are very
comfortable with algebra.
Which Mathematical Practices are Highlighted in Solving the
Consecutive Natural Numbers Task?
Each Standard for Mathematical Practice has many aspects. The multiple features of
each practice are part of what makes the practices worth teaching. However, any
particular task is likely to highlight only a few of the ways that each practice could come
up. Here we summarize the key aspects of each practice that could be highlighted by
the Consecutive Natural Number Task. Each phrase or sentence listed as a
mathematical practice is taken from JR’s Highlights of the SMP for Rich Tasks
document that can be downloaded from the MP section of the website.
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JR Task Analysis
Sum of Consecutive Natural Numbers
MP1 Make sense of problems and persevere in solving them.
A major purpose of the Consecutive Natural Numbers Task (CNNT) is generalizing from
numeric examples to statements of conjectures that can be proven. This purpose may
not be initially apparent, but it becomes apparent after teachers wonder if there is a
better way to analyze the numbers from 1 to 49 than by exploring each number
separately. Thus, making sense of the task involves finding entry points to solving it.
These entry points are conjectures that, once proven, allow teachers to eliminate
various numbers from consideration. One way to arrive at conjectures is to try smaller
numbers of addends before larger numbers of addends; this simpler version of the
problem yields reasoning strategies that can be used for a greater numbers of addends
(e.g., going from three addends to five or seven and then any odd number of addends).
Before examining patterns, teachers need to make sense of the statement of the task: it
is asking two things, for those numbers that can and cannot be written as the sum of
consecutive natural numbers. Finding the numbers that are cannot requires identifying
the ones that can be expressed as a sum of consecutive natural numbers. Furthermore,
to show that a number cannot be written as a sum of consecutive natural numbers
means eliminating the possibility that any sum of consecutive natural numbers would
equal that number.
Persevering through the task involves phrasing conjectures as precisely and generally
as possible. Phrasing such conjectures involves explaining correspondences among
numerical expressions, verbal descriptions and algebraic expressions. Checking that a
phrasing is correct and making revisions as necessary requires asking, "Does this make
sense?" and comparing the statement to tests of the conjecture.
Very Important
Somewhat Important
• Make sense of the problem.
• Analyze givens and constraints.
• Look for entry points.
• Calculate accurately and efficiently.
• Make conjectures about possible
solution strategies.
• Determine whether the answers
make sense.
• Understand the approaches of
others to solving problems.
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JR Task Analysis
Sum of Consecutive Natural Numbers
MP2 Reason abstractly and quantitatively.
Algebra is often the language used to abstract ideas and communicate them. Key work
involves turning observations (“a sum of two consecutive natural numbers is odd”) into
algebraic statements (𝑥 + [𝑥 + 1] = 2𝑥 + 1).
Very Important
Somewhat Important
• Create a coherent representation of the
problem.
• Attend to the meaning of quantities.
• Look for entry points.
• Know and flexibly use different
properties of operations.
• Make conjectures about possible
solution strategies.
MP3 Construct viable arguments and critique the reasoning of others.
Here the work involves looking at various cases (sums of chains of 2, 3, 4, etc,
consecutive natural numbers) for generating a rule that works with any set of
consecutive natural numbers of a given length. Given that there are many different rules
that could work, listening to and understanding of approaches presented by colleagues
is essential. In this task we persevered in exploring the task even if a solution (i.e., a
proof about the powers of two) is not reached.
As usual, participants construct arguments based on numerical examples. They make
conjectures and construct a sequence of statements to support their conjectures. They
also are able to compare the effectiveness of two plausible arguments by listening to
others’ explanations. Participants can extend their reasoning with numerical examples
to using symbols to generalize their thinking. The question here is “Does that work for
any sum of 𝑥 consecutive natural numbers?”
Very Important
Somewhat Important
• Make conjectures and explore them.
• Disprove conjectures with counter
examples.
• Support conjectures with examples.
• Construct logical arguments.
• Use stated assumptions, definitions,
and previously established results.
• Analyze situations by breaking them
into cases.
• Justify conclusions, and
communicate them to others.
• Ask useful questions.
CNNT
• Respond to feedback.
• Evaluate the spoken or written
reasoning of others.
• Compare the effectiveness of
plausible arguments
• Distinguish between correct and
flawed reasoning.
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JR Task Analysis
Sum of Consecutive Natural Numbers
MP6 Attend to precision.
Precision in communication is essential as is the introduction of symbols to generalize
their results. Most importantly, definitions of consecutive natural numbers, odd and even
numbers, and multiples of a number form the basis of the reasoning upon which the
proofs are built.
Very Important
Somewhat Important
• Communicate precisely to others.
• Use the equal sign consistently and
appropriately.
• Use clear definitions in discussions with
others and in their own reasoning.
• Calculate accurately and efficiently.
• State the meaning of the symbols used.
• Give carefully formulated explanations
to others.
MP7 Look for and make use of structure.
Key work in this problem involves looking for the structure in a sequence of numbers of
a particular length. Specifically, the numbers in the sequences investigated differ by 1,
are equally spaced, and are centered on a middle/median value. The use of the
structure of a sequence of numbers enables participants to generalize their findings and
“prove” them.
Very Important
Somewhat Important
• Look closely to see the structure of
mathematics.
* Step back for an overview and shift
perspective if necessary
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JR Task Analysis
Sum of Consecutive Natural Numbers
MP8 Look for and express regularity in repeated reasoning.
The regularity of a sequence of consecutive natural numbers of a given length enables
participants to move from specific examples to a generalized formulation. This can lead
to the proofs of several conjectures.
Very Important
Somewhat Important
• Look out for patterns.
• Notice when calculations [or reasoning]
is repeated, and if so, look for both
generalized methods and shortcuts.
• Continually evaluate the
reasonableness of the intermediate
results.
Which CCSS Grade Level Content Standards Could the
Consecutive Natural Numbers Task Address?
Operations and Algebraic Thinking 2.OA
Represent and solve problems involving addition and subtraction.
3. Determine whether a group of objects (up to 20) has an odd or even number of
numbers, e.g., by pairing objects or counting them by 2s; write an equation to
express an even number as a sum of two equal
Solve problems involving the four operations, and identify and explain
patterns in arithmetic.
9. Identify arithmetic patterns (including patterns in the addition table or
multiplication table), and explain them using properties of operations. For
example, observe that 4 times a number is always even, and explain why 4 times
a number can be decomposed into two equal addends.
Operations and Algebraic Thinking 4.OA
Gain familiarity with factors and multiples.
4. Find all factor pairs for a whole number in the range 1–100. Recognize that a
whole number is a multiple of each of its factors. Determine whether a given
whole number in the range 1–100 is a multiple of a given one-digit number.
Determine whether a given whole number in the range 1–100 is prime or
composite.
CNNT
Jacobs, Kennedy, Lai & Sherman
http://www.judithrules.com
p. 14/15
JR Task Analysis
Sum of Consecutive Natural Numbers
Generate and analyze patterns.
5. Generate a number or shape pattern that follows a given rule. Identify apparent
features of the pattern that were not explicit in the rule itself. For example, given
the rule “Add 3” and the starting number 1, generate terms in the resulting
sequence and observe that the terms appear to alternate between odd and even
numbers. Explain informally why the numbers will continue to alternate in this
way.
Statistics and Probability 6.SP
Develop understanding of statistical variability.
3. Recognize that a measure of center for a numerical data set summarizes all of its
values with a single number, while a measure of variation describes how its
values vary with a single number.
Extensions
1. What happens if you are allowed to use 0 as one of the numbers less than 50?
Everything else about the problem stays the same. Make some conjectures and
try to explain why they make sense mathematically.
2. What happens if we look at the range of numbers from −20 to +20? Again,
everything else about the problem stays the same, Make some conjectures and
explain why they make sense mathematically.
3. What about the sums of consecutive even natural numbers? What conjectures
can you make about those sums?
4. What about the sums of consecutive odd natural numbers? What conjectures
can you make about those sums? Explain why they make sense mathematically
5. Prove: Any natural number of the form 2! cannot be written as the sum of
consecutive natural numbers.
CNNT
Jacobs, Kennedy, Lai & Sherman
http://www.judithrules.com
p. 15/15