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? [2]
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]
6. 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 = +
7. 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]
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]
Section E: Multiples of Odd Numbers
15. 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 + _____ + _____ + _____ + _____ + _____
16. 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. [1]
Final Score:
/ 30
Teacher’s Comments (if any):
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. [1]
Because the systematic list (or pattern) in Q1 shows that there is only 1 way. [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. [1]
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]
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.]
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 [1]
13. 27 = 13 + 14; 27 = 8 + 9 + 10; 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__
16. [1 mark if all entries (in bold) correct; 0 mark if minor mistakes]
7 = 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
17. 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.]
19. Some suggestions for main lesson learnt from this worksheet: [1]
ï I have learnt which numbers can be written as the sum of consecutive natural numbers.
ï I have learnt how to carry out mathematical investigation on my own (see Q11).
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