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Whole number and polygon

Task1
some whole numbers can be written as the sum of consecutive whole numbers.
for example: 5=2+3 12=3+4+5
some whole numbers can be written as the sum of consecutive whole numbers in more than one way. for example: 9=4+5=2+3+5

which whole numbers can be written as the sum of consecutive whole numbers? can they all? if we have a specific whole number, how can i tell whether it can be written as the sum of consecutive whole numbers or not and if so in how many different ways? How would you justify your response?

Task2
if we have a polygon with n sides, how many diagonals will it have? why? How could you convince someone who was not sure?

1- after that work on both mathematical tasks yourself and provide a detailed written account of the work on each of them, including both what you found out and how you went about tackling the task. Where did you get stuck and what did you do about it? Which of the seven curriculum mathematical processes (communication, connection, reasoning, technology, visualization,problem solving, or estimation and mental mathematics) did you use and how?

2-Then select one of these tasks and analyze it in terms of its mathematical demand. what did you need to know mathematically in order to be able to work on it/ solve it as you did? Then decide which grade(s) in either junior or senior high school your chosen task would be appropriate for. Be explicit and thorough in your rationale.

3- How would you modify the task you chose in order to make it usable at grade 7 and grade 12? justify your response (even if you judge no modification is required) in terms of both the mathematic features and demands of the task and the program of studies.

Solution Preview

Please see the attached file.

Task1
some whole numbers can be written as the sum of consecutive whole numbers.
for example: 5=2+3 12=3+4+5
some whole numbers can be written as the sum of consecutive whole numbers in more than one way. for example: 9=4+5=2+3+5

which whole numbers can be written as the sum of consecutive whole numbers? can they all? if we have a specific whole number, how can i tell whether it can be written as the sum of consecutive whole numbers or not and if so in how many different ways?
How would you justify your response?

The concept we are discussing here is sum of run of numbers. The definition of run of number is as below. A run of numbers is a sequence of consecutive whole numbers i.e. with no gaps in the sequence,
Such as 2,3,4, is a run of numbers
And 7, is also a run even though it contains just a single number. However, this run is called trivial run. A run with more than one number is called an interesting run.
But not 5,6,8,9 because 7 is missing
And not 2,3,3 because 3 is repeated.

Every number is therefore a sum of numbers in a run because we can have a run with just a single number in it!

Many numbers can also be written as a sum of a run of 2 or more numbers. For example:
9 = 2 + 3 + 4 = 4 + 5
10 = 1 + 2 + 3 + 4
11 = 5 + 6
12 = 3 + 4 + 5
13 = 6 + 7
14 = 2 + 3 + 4 + 5
For convenience, the sum of a run of numbers we will call a runsum.
So how can we tell whether there exist an interesting run for a number and how many ways we can perform it?
For example, we want to find out whether 36 can be written as the sum of two, three, four or five consecutive numbers. Use the following table to record information.

Sum of 2 consecutive numbers Sum of 3 consecutive numbers Sum of 4 consecutive numbers Sum of 5 consecutive numbers
3 6 10 15
5 9 14 20
7 12 18 25
9 15 22 30
11 18 26 35
13 21 30 40

From this table, we see some interesting observations:
• the sums of two consecutive numbers are all odd.
• the sums of three consecutive numbers ...

Solution Summary

The solution provides a detailed and step-by-step explanation (4-page Word file) for the problem.

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