Intent The purpose of this activity is not for students to learn a proof that odd numbers greater than 1 can be written as a sum of two consecutive numbers. Rather, it is intended to help students begin to learn what a proof is learn to distinguish between specific examples and a general argument gain experience in communicating complex, abstract ideas become familiar with a more precise way of thinking than they may have encountered before

Mathematics This activity challenges students to evaluate whether a conjecture is true or false.The conjecture in question is “If an odd number is greater than 1, then it can be written as the sum of two consecutive numbers.” Students are asked to find a counterexample if they think the statement is false or to devise a set of instructions for writing any odd number greater than 1 as a sum of two consecutive numbers if they think the statement is true.

Progression This activity should follow the conclusion of Consecutive Sums. Students work alone to consider a conjecture likely to have been made during that activity. Follow-up discussion will introduce the concept of a proof.

Approximate Time 15 minutes for activity (at home or in class) 10 minutes for discussion

Classroom Organization Individuals, followed by whole-class discussion

Doing the Activity Introduce the activity by asking, Is the conjecture in the activity true? How confident are you about your answer? Clarify the instructions for students so that they understand what is expected of them.

Discussing and Debriefing the Activity Begin the discussion by asking the class again whether they think the conjecture is true. Then ask how confident they are that it is true for every odd number greater than 1. Most students will likely be fairly sure that it is always true, but encourage skeptics to voice their opinions. Ask for volunteers to share any instructions they developed for writing any odd number as a sum of two consecutive numbers, and have them illustrate their methods using specific examples. If the class is at a loss about how to do this, you might ask a series of questions, such as, How would you write 397 as the sum of two consecutive numbers? How would you write 4913 as the sum of two consecutive numbers? How would you write 157,681 as the sum of two consecutive numbers? Encourage students to explain how to find the pair of consecutive integers in each case, as this is key to developing a general argument. There are several ways to describe the general process; elicit as many as possible from your students. Here are some commonly suggested procedures. Subtract 1 from the odd number to get an even number. Divide this even number by 2. That quotient and the next number are the desired consecutive numbers. Divide the odd number by 2, getting “something and a half.” The whole numbers just above and below this mixed number are the desired consecutive numbers. Add 1 to the odd number to get an even number. Divide this even number by 2. That quotient and the previous number are the desired consecutive numbers. Careful examination of any of these methods will show that they don’t work if the initial odd number is 1, because one of the numbers in the consecutive sum will be 0 rather than a positive whole number as required. Whichever methods students suggest, ask them to explain how they know that a given method works.For example, for the first procedure above, you might ask how students know that subtracting 1 from an odd number gives an even number. The best response to this question would refer to a definition of the term odd. That is, students should recognize that, ultimately, they can’t say anything for sure about odd numbers unless they begin with a clear definition. Similarly, ask how students know that dividing an even number by 2 gives a whole-number result. Again, encourage them to see that the answer to this challenge depends on having a precise definition of the term even. It is not necessary to go into formalities about the meaning of the terms odd and even. What is important is recognizing the value of having a precise definition if one is to give a complete proof. Use the discussion to help bring out the difference between a collection of examples of a phenomenon and a legitimate general proof. A proof does not need to use algebraic symbols. For example, when appropriate and precise definitions are given for odd and even, the arguments above constitute completely legitimate proofs that every odd number can be written as a consecutive sum with two terms. Help students to see that these arguments are better than only giving a few examples such as 23 = 11 + 12 and 47 = 23 + 24. Each procedure listed above demonstrates that every odd number is expressible as a consecutive sum of two terms by showing how to do it, that is, how to find the two terms. Such how-to arguments are considered legitimate proofs and are known as constructive proofs. Algebraic symbols do sometimes help students understand a situation, and your students may be able to express their arguments symbolically. For example, if you suggest using n for the number obtained after subtracting 1 and dividing by 2, students can probably write the next number as n + 1. You might extend this problem by encouraging students to express each method using a general equation. For instance, the first method listed above can be expressed by the equation N=N−12+N−12+1 size 12{N= { {N-1} over {2} } + left ( { {N-1} over {2} } +1 right )} {}. Students can be asked to explain why, if N is odd and greater than 1, both terms must be positive integers.

Key Questions Is the conjecture in the activity true? How confident are you about your answer? How would you write 397 as the sum of two consecutive numbers? How would you write 4913 as the sum of two consecutive numbers? How would you write 157,681 as the sum of two consecutive numbers?

Supplemental Activities The General Theory of Consecutive Sums (extension), the final activity in this group, asks students to explore consecutive sums of integers. Infinite Proof (extension) asks students to prove that the square of every odd number is odd and that every prime number greater than 10 must have 1, 3, 7, or 9 as its units digit.