One class period to launch, intermittent over 6-10 days, with time at the end for the solution.
Ohio Standards Alignment
- Prime factorization
- Proof by contradiction
Arithmetic sequences and series, sigma notation, prime factorization, proof by contradiction
This problem is intended to be given during a unit in which arithmetic sequences and series, formulas, and sigma notation are being developed. The purpose is to put the student in the role of mathematical investigator. The problem can be posed at the beginning of the unit and conducted largely as an individual or pair investigation outside of class. Students investigate series of consecutive whole numbers with the aim of answering these questions: Can any whole number greater than 2 be written as a sum of consecutive whole numbers? If not, which can (cannot)? The problem is constructed in a sequence of easy-medium-difficult stages designed to acquaint the solver with the processes of mathematics: listing specific numerical examples, searching for patterns, generalizing, writing definitions, constructing symbols, and proving results. Four related questions are posed, with accompanying hints to be given as needed to prompt the solver toward a productive solution path. The ideas involved in the investigation utilize the same principle of nested pairs of numbers about the midpoint of an arithmetic sequence that leads to the formula for the sum of an arithmetic series. In the course of the investigation, students must practice the ideas that lead up to this important result, as well as discover mathematical ideas on their own and use the mathematical processes identified above.
- To conduct a mathematical investigation of sequences of consecutive whole numbers.
- To construct multiple decompositions for numbers that can be written as the sum of consecutive whole numbers.
- To develop an algorithm for finding such sums.
- To prove that powers of 2 cannot be written as such a sum.
- To practice through numerical examples the nesting ideas involved in developing the formula for sums of finite arithmetical series.
Some whole numbers can be written as a sum of consecutive whole numbers. Others cannot. Some can be written as more than one such sum. Here are some examples:
3 = 1+2
4 cannot be written as such a sum
6 = 1+2+3
15 = 1+2+3+4+5 or 15 = 7+8 or 15 = 4+5+6
Given any whole number greater than 2, can you determine whether it can or cannot be written as a sum of consecutive whole numbers? If so, can you write one or more sums that equal that number?
The investigation is laid out step by step on a student activity sheet. The basic questions are given here:
- Which whole numbers greater than 2 can be written as a sum of consecutive whole numbers and which cannot?
- For numbers that can be written as a sum of consecutive whole numbers, (a) describe a written process for finding and writing that sum; (b) develop an algorithm for determining whether a sum exists for any given number and for finding such a sum if it exists.
- For certain numbers that can be written as more than one sum of consecutive whole numbers, describe a procedure for finding three or more such sums for the given number.
- Develop a proof that certain whole numbers cannot be written as a sum of consecutive whole numbers.
About the time required for this problem: The intent is that about one class period be used to launch the initial investigation. However, it takes a considerable amount of time outside of class for students to play around with the numbers to see how series actually behave. This problem does not require so much time at any given moment, but it requires returning to time and again.
In the interim, students need to be exploring related ideas during their regular lessons, such as formulas for summing series. These ideas are required for some components of the rich problem, and in particular, for a proof of the result (which I would expect few students to actually attempt). However, the proof is something which almost all students can profit from when it is gone over in class.
Therefore, the nature of this lesson makes it difficult to assign a time required. In listing the estimated time as 6-10 class periods, I mean the "time span" over which the problem merits consideration. This time span, alone, will be a novel feature of the problem, since high school students are not often accustomed to working on something more than one or two class periods. Hence, the hints and varying amounts of credit that can be earned will help students individually choose just how far to pursue the problem.
Further discussion, examples, and a complete proof of the result can be found on the Solutions sheet.
From the teaching files of Steven P. Meiring.