Mathematical Investigation: Polite Numbers
Topic: Numbers
The purpose of this worksheet is to investigate when a natural number can be written as the sum of
consecutive natural numbers.
Section A: Sum of Two Consecutive Natural Numbers
1.
The odd number 9 can be written as the sum of two consecutive natural numbers: 9 = 4 + 5.
Express the following odd numbers as the sum of two consecutive natural numbers. If it is not
possible to do so, indicate ‘Not possible’ beside the number.
[1]
1 =
+
3 =
+
5 =
+
7 =
+
9 = 4 + 5
11 =
+
2.
What pattern(s) do you notice in Q1?
3.
Using the pattern(s) you identify in Q2, express 2007 as the sum of two consecutive natural
numbers.
[1]
4.
Do you think all odd numbers, except 1, can be written as the sum of two consecutive natural
numbers? Why or why not?
[2]
5.
How many ways can the number 9 be written as the sum of two consecutive natural numbers?
Why?
[2]
© Joseph Yeo
[2]
1
6.
7.
Express the following even numbers as the sum of two consecutive natural numbers. If it is
not possible to do so, indicate ‘Not possible’ beside the number.
[1]
2 =
+
4 =
+
6 =
+
8 =
+
10 =
+
Do you think all even numbers can be written as the sum of two consecutive natural numbers?
Why or why not?
[2]
Section B: Sum of Three Consecutive Natural Numbers
8.
The number 9 can be written as the sum of three consecutive natural numbers: 9 = 2 + 3 + 4.
Express the following numbers as the sum of three consecutive natural numbers. If it is not
possible to do so, indicate ‘Not possible’ beside the number.
[2]
1 =
+
+
2 =
+
+
3 =
+
+
4 =
+
+
5 =
+
+
6 =
+
+
7 =
+
+
8 =
+
+
9 =
+
+
9.
By looking at the pattern in Q7, do you think which numbers can be written as the sum of
three consecutive natural numbers? Why?
[2]
10.
How many ways can the number 9 be written as the sum of three consecutive natural
numbers? Why?
[1]
© Joseph Yeo
2
Section C: Sum of Four Consecutive Natural Numbers
11.
Which numbers can be written as the sum of four consecutive natural numbers? Investigate.
Try to explain why it happens.
[4]
Section D: Sum of Consecutive Natural Numbers
12.
The number 9 can be written as the sum of consecutive natural numbers in exactly two
different ways (order of the consecutive natural numbers is not important):
9
= 4 + 5
9
= 2 + 3 + 4
Express the number 18 as the sum of consecutive natural numbers in exactly two different
ways.
[1]
13.
Find a number that can be written in exactly three different ways and show the three different
sums.
[1]
14.
Find a number that can be written in exactly four different ways and show the four different
sums.
[1]
© Joseph Yeo
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Section E: Multiples of Odd Numbers
15.
16.
The multiples of 9 are 9, 18, 27, 36, 45, … We will try to express the multiples of 9 as the
sum of consecutive natural numbers. Continue the pattern below.
[1]
9 =
4 + 5
18 =
3 + 4 + 5 + 6
27 =
_____ + 3 + 4 + 5 + 6 + _____
36 =
_____ + _____ + 3 + 4 + 5 + 6 + _____ + _____
45 =
_____ + _____ + 3 + 4 + 5 + 6 + _____ + _____ + _____
54 =
_____ + 3 + 4 + 5 + 6 + _____ + _____ + _____ + _____
63 =
3 + 4 + 5 + 6 + _____ + _____ + _____ + _____ + _____
72 =
4 + 5 + 6 + 7 + _____ + _____ + _____ + _____ + _____
81 =
5 + 6 + 7 + 8 + _____ + _____ + _____ + _____ + _____
Using the same pattern as above, express the first six multiples of 7 as the sum of consecutive
natural numbers.
[1]
7 =
14 =
21 =
28 =
35 =
42 =
17.
Do you think all multiples of odd numbers (other than multiples of 1) can be written as the
sum of consecutive natural numbers? Why or why not?
[2]
Conclusion
18.
Which numbers cannot be written as the sum of consecutive natural numbers? By looking at
all the previous questions, write down the first 6 numbers that cannot be written as the sum of
consecutive natural numbers. What do you notice about these numbers?
[2]
19.
Write down one main lesson that you have learnt from this worksheet.
© Joseph Yeo
4
[1]
Final Score:
/ 30
Final Score
24-30
20-23
15-19
10-14
0-9
Grade
A
B
C
D
F
Teacher’s Comments (if any):
© Joseph Yeo
5
Mathematical Investigation: Polite Numbers
Topic: Numbers
Answers & Scoring Rubric
(Marks allocated in square brackets; total marks = 30)
Section A: Sum of Two Consecutive Natural Numbers
1.
[1 mark if all entries (in bold) correct; 0 mark if minor mistakes]
1 =
Not Possible
3 =
1 + 2
5 =
2 + 3
7 =
3 + 4
9 = 4 + 5
11 =
5 + 6
2.
Pattern 1:
Both consecutive numbers increase by 1 as you go down to the next row. In other
words, the next row after 5 + 6 will be 6 + 7 = 13.
[1]
Pattern 2: Divide the odd number by 2. Then subtract half from the answer to get the first
consecutive number, and ass half to the answer to get the second consecutive
number. E.g. 7  2 = 3.5. So the two consecutive numbers are 3.5 – 0.5 = 3 and
3.5 + 0.5 = 4.
[1]
[Or any other patterns]
3.
From Pattern 2, 2007  2 = 1003.5. Therefore 2007 = 1003 + 1004.
[1]
[If the student did not use any pattern but do it by brute force, or guess and check, it is up to
you whether to award the mark. For weaker students, maybe you can award the mark.]
4.
Yes, all odd numbers, except 1, can be written as the sum of two consecutive natural numbers.
[1]
Because we can always use Pattern 2 to do it.
[1]
[Note for the Teacher (your students may not understand this): All odd numbers are of the
form 2n + 1, where n = 0, 1, 2, … But 2n + 1 = n + (n + 1), which is the sum of two
consecutive numbers. But we want n to be a natural number, so n > 0 and this excludes the
odd number 1 where n = 0.]
5.
Only 1 way.
Because the systematic list (or pattern) in Q1 shows that there is only 1 way.
[1]
[1]
[Or the only way to change the two consecutive numbers is to add or subtract the same
number from each of these numbers. E.g. if you subtract 1 from each of the two consecutive
numbers, 4 and 5, then you will get 3 + 4 which is less than 9; if you add 1 to each of the two
consecutive numbers, 4 and 5, then you will get 5 + 6 which is more than 9. Since you have to
either add or subtract to get two other consecutive numbers, then you will never get the same
value of 9.]
6.
All not possible.
7.
No, all even numbers cannot be written as the sum of two consecutive natural numbers. [1]
Because if you have two consecutive natural numbers, one of them must be odd and the other
even. But the sum of an odd number and an even number will always be odd.
[1]
© Joseph Yeo
[1]
6
Section B: Sum of Three Consecutive Natural Numbers
8.
[2 marks if all entries (in bold) correct; 1 mark if minor mistakes; 0 mark if major mistakes]
6 =
1 + 2 + 3
9 =
2 + 3 + 4
Not possible for the other numbers.
9.
All multiples of 3, except 3, can be written as the sum of three consecutive natural numbers.
[1]
[Accept: 6, 9, 12, … since students have not learnt multiples of 3 at this stage.]
Because when you go to the next row, each of the three consecutive natural numbers will
increase by 1:
6 = 1 + 2 + 3
9
=
2 + 3 + 4
Therefore, the next number that can be written as the sum of three consecutive natural
numbers will increase by 3. Since the first number is a multiple of 3, then the rest will be
multiples of 3.
[1]
[This second part may be too difficult for weaker students to infer on their own. But they
may be able to understand if you explain to them.]
10.
Only 1 way because the systematic list (or pattern) in Q8 shows that there is only 1 way. [1]
[For another way of explaining, see comments at end of Q5.]
Section C: Sum of Four Consecutive Natural Numbers
11.
Generating specific examples:
[2]
Method 1: Trial and error by looking at all the natural numbers starting from 1 (similar to
Q7)
Method 2: Start by writing:
1 + 2 + 3 + 4 = 10
2 + 3 + 4 + 5 = 14
3 + 4 + 5 + 6 = 18
[2 marks if use Method 2 because it shows that students have observed this pattern from Q1
and Q7; 1 mark if use Method 1; for any other method, you decide.]
Generalisation:
[1]
Natural numbers that can be written as the sum of four consecutive natural numbers are 10,
14, 18, …
[That is, the common difference between successive terms is 4. But unlike Q8, the first term
is not a multiple of 4 and so these are not multiples of 4.]
© Joseph Yeo
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Why the pattern occurs:
[1]
When you go to the next row, each of the four consecutive natural numbers will increase by 1.
Therefore, the next number that can be written as the sum of four consecutive natural numbers
will increase by 4. This explains why the (common) difference between successive terms is 4.
[In fact, the pattern is that the common difference, n, between successive terms is the same as
the sum of n consecutive natural numbers. E.g. if you want to express natural numbers as the
sum of 5 consecutive natural numbers, then the common difference between these natural
numbers is 5. Whether they are multiples of 5 will depend on whether the first natural number
is a multiple of 5 or not. In this case, the answer is yes, since 1 + 2 + 3 + 4 + 5 = 15.]
Section D: Sum of Consecutive Natural Numbers
12.
18 = 5 + 6 + 7;
18 = 3 + 4 + 5 + 6
13.
27 = 13 + 14;
27 = 8 + 9 + 10;
[1]
27 = 2 + 3 + 4 + 5 + 6 + 7
[1]
[If you really want to prove that there are only 3 different ways:
 From Q5 and Q10, there is only one way to express 27 as the sum of two consecutive
natural numbers, or as the sum of three consecutive natural numbers, or as the sum of six
consecutive natural numbers.
 27 cannot be written as the sum of four consecutive natural numbers because this sum
will always be even: odd + even + odd + even = even.
 27 cannot be written as the sum of five consecutive natural numbers because this sum will
always be multiples of 5: see comments at end of Q11.
 For the sum of seven or more consecutive natural numbers, all these sums will be greater
than 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28 which is more than 27 already.]
14.
63 = 31 + 32;
63 = 20 + 21 + 22;
63 = 6 + 7 + 8 + 9 + 10 + 11 + 12;
63 = 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11
[1]
[If you really want to prove that there are only 4 different ways, see similar argument at end
of Q13.]
Section E: Multiples of Odd Numbers
15.
[1 mark if all entries (in bold) correct; 0 mark if minor mistakes]
9 =
4 + 5
18 =
3 + 4 + 5 + 6
27 =
__2__ + 3 + 4 + 5 + 6 + __7__
36 =
__1__ + __2__ + 3 + 4 + 5 + 6 + __7__ + __8__
45 =
__1__ + __2__ + 3 + 4 + 5 + 6 + __7__ + __8__ + __9__
54 =
__1__ + 3 + 4 + 5 + 6 + __7__ + __8__ + __9__ + __10__
63 =
3 + 4 + 5 + 6 + __7__ + __8__ + __9__ + __10__ + __11__
72 =
4 + 5 + 6 + 7 + __8__ + __9__ + __10__ + __11__ + __12__
81 =
5 + 6 + 7 + 8 + __9__ + __10__ + __11__ + __12__ + __13__
© Joseph Yeo
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16.
[1 mark if all entries (in bold) correct; 0 mark if minor mistakes]
7
17.
=
3 + 4
14 =
2 + 3 + 4 + 5
21 =
1 + 2 + 3 + 4 + 5 + 6
28 =
1 + 2 + 3 + 4 + 5 + 6 + 7
35 =
2 + 3 + 4 + 5 + 6 + 7 + 8
42 =
3 + 4 + 5 + 6 + 7 + 8 + 9
Yes, all multiples of odd numbers (other than multiples of 1) can be written as the sum of
consecutive natural numbers.
[1]
Because we can use the pattern in Q15 or Q16 to generate them.
[1]
Conclusion
18.
The first 6 numbers that cannot be written as the sum of consecutive natural numbers are:
1, 2, 4, 8, 16, 32
[1]
They are powers of 2 (where the power is a whole number).
[1]
[Accept similar answers because students have not learnt powers of 2.]
Notes for the Teacher:
 The problem is “Which number can or cannot be expressed as the sum of consecutive
natural numbers?”
 All odd numbers, except 1, can be expressed as the sum of consecutive natural numbers (see
Q1-4).
 The problem is the even numbers. All even natural numbers can be classified into two
categories: multiples of odd numbers (excluding multiples of 1) and powers of 2.
 If you divide an even number by 2 and if the resulting number is not an odd number, then
divide the resulting number by 2 again. If you continue this process, you will eventually get
an odd number n. If the odd number n is not 1, then the original even number will be a
multiple of n. If the odd number n is 1, then the original even number is a power of 2.
 Even numbers, which are multiples of odd numbers (excluding multiples of 1), can be
expressed as the sum of consecutive natural numbers (see Q15-17).
 Even numbers, which are powers of 2 where the power is a whole number, cannot be
expressed as the sum of consecutive natural numbers (see Q18). The proof is beyond the
students but if you are interested, please download it from http://math.nie.edu.sg/bwjyeo.
19.
Some suggestions for main lesson learnt from this worksheet:
[1]
 I have learnt which numbers can be written as the sum of consecutive natural numbers.
 I have learnt how to carry out mathematical investigation on my own (see Q11).
© Joseph Yeo
9
Students’ Samples
Another version of this problem was given to a few classes of Sec 1 students. Because these students
are more mathematically inclined, the investigation was more open-ended with very little guidance.
The original problem given to them is printed below:
The number 9 can be written as the sum of consecutive natural numbers in two different ways:
9=2+3+4
9=4+5
The number 27 can be written as the sum of consecutive natural numbers in three different ways:
27 = 13 + 14
27 = 8 + 9 + 10
27 = 2 + 3 + 4 + 5 + 6 + 7
Look at numbers other than 9 and 27 and find out all you can about writing them as sums of
consecutive natural numbers.
(a) Which numbers cannot be written as consecutive sums?
(b) What kinds of numbers can be written as the sum of 2 or 3 or 4 or more consecutive numbers?
(c) How many ways can various numbers be written as?
Write about your discoveries. Try to explain why they occur.
Some Students’ Discoveries (that is also in this mathematical investigation)

Numbers that can be written as the sum of two consecutive natural numbers are all odd numbers
except 1.
Examples: 1 + 2 = 3, 2 + 3 = 5, … (see Q1-4)

Numbers that can be written as the sum of three consecutive natural numbers are all multiples of 3
except 3.
Examples: 1 + 2 + 3 = 6, 2 + 3 + 4 = 9, … (see Q8-9)

Numbers that can be written as the sum of four consecutive numbers are all even numbers, or the
different between successive terms is 4.
Examples: 1 + 2 + 3 + 4 = 10, 2 + 3 + 4 + 5 = 14, … (see Q11)

Multiples of odd numbers (other than multiples of 1) can be written as the sum of consecutive
natural numbers (see Q15-17)

2n, where n  0, cannot be written as the sum of consecutive natural numbers (see Q18)
Some Students’ Discoveries (that is NOT in this mathematical investigation)

All prime numbers (except 2) can be written as the sum of two consecutive natural numbers. This
is because all prime numbers (except 2) are odd numbers other than 1 (see Q1-4).

n2 where n is odd (or perfect squares that are odd) can be written as the sum of consecutive natural
numbers. This is because these are multiples of odd numbers except multiples of 1 (see Q15-17).

If the number n can be written as the sum of consecutive natural numbers, then 2n can also be
written as the sum of consecutive natural numbers. This is because n can either be an odd number
other than 1, or a multiple of an odd number m (other than a multiple of 1) if n can be written as
the sum of consecutive natural numbers (see comments at end of Q18). Therefore, 2n is just a
© Joseph Yeo
10
multiple of the odd number n or a multiple of the odd number m and so it can be written as the
sum of consecutive natural numbers (see Q15-17).
[In fact, 3n, 4n, 5n , etc., can also be written as the sum of consecutive natural numbers since they
are just multiples of the odd number n or m.]

Number pattern of numbers that can be written as the sum of consecutive natural numbers:
Sum of two consecutive natural numbers:
Sum of three consecutive natural numbers:
Sum of four consecutive natural numbers:
Sum of five consecutive natural numbers:
Sum of six consecutive natural numbers:

All odd numbers produced
Numbers produced are odd, even, odd, even, …
All even numbers produced
Numbers produced are even, odd, even, odd, …
All odd numbers produced
The bigger the number, the more ways it can be written as the sum of consecutive natural
numbers. This looks reasonable but it is not true. For example, 15 can be written as the sum of
consecutive natural numbers in exactly 3 ways but 62 can be written as the sum of consecutive
natural numbers in only 1 way.
Authors’ Comments

Please note that the unguided version was given to the students after they have covered topics on
multiples, prime numbers and number patterns. So they have more things to say.

Students who discovered the above patterns usually did not prove them or explain why the
patterns occur. This is partly because it is beyond their capability. However if you use Alternative
Assessment 1 which is more guided, it is still possible to guide your students to explain why the
patterns occur. The ability to discover a pattern is important, but to be able to explain why it
occurs requires even higher order thinking skills.

In fact, some students generalised too early. It just happens that some of their generalisations are
true but some of them are wrong.

For odd prime numbers, it seems that there is only one way to express it as the sum of consecutive
natural numbers (namely as the sum of two consecutive natural numbers because they are all
odd).

You can also use an Excel spreadsheet to help in the investigation. For more info, download the
file PoliteNumbers.xls from http://math.nie.edu.sg/bwjyeo. Click on each cell and see the formula
in the toolbar on top. There is a simple way to auto-complete the formulae for all the
corresponding cells, or else it will be tedious to key in so many formulae.
© Joseph Yeo
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