Proof inquiry
The prompt
Mathematical inquiry processes: Explore, identify and explain patterns, generalise, and prove. Conceptual field of inquiry: Factors and multiples, terms and expressions, algebraic proof.
Derek Christensen, a teacher of mathematics in grades 10 to 12 at W. P. Wagner High School in Edmonton (Canada), devised the prompt. He reports that it is good for moving students from inductive thinking to deductive reasoning.
As Polya writes in How to Solve It, "Mathematics presented with rigour is a systematic deductive science but mathematics in the making is an experimental inductive science."
The prompt unites the two sides of the discipline and, through inquiry, students learn that the inductive side on its own is insufficient to show a generalisation is always true.
In the question, notice, and wonder phase of the inquiry, students have offered definitions for sum, consecutive, integer, and multiple before giving examples, making conjectures, and wondering about other cases:
The sum of the first three consecutive even integers (two, four, and six) is 12 and that is a multiple of six.
14 + 16 + 18 = 48 which is a multiple of six.
All the examples we've done lead to a multiple of six so it is always true.
Can you use negative integers?
Is it always true for three consecutive odd integers?
Does it work for any three even integers?
Is it a multiple of six because there are three integers?
Induction to deduction
The teacher can help students move from inductive exploration to deductive reasoning by using the form of algebraic proof when writing specific examples. For example, 14 + 16 + 18 = 48 = 6(8) prepares the ground for 2n + (2n + 2) + (2n + 4) = 6n + 6 = 6(n + 1)
Even though the proof makes the prompt true for all values of n (except n = 0), some students can remain unconvinced. They want to verify its truth by substituting values of n into the first part of the proof - in effect, going from the general to the particular.
This is not 'wrong'. After all, a mathematician might test a general approach with a particular case. However, it does suggest the student has not understood the universal nature and completeness of the proof.
Generating further inquiry
For a class new to inquiry, the teacher might plan for a structured inquiry in which students follow the lines of inquiry below one after the other.
However, students should be encouraged to develop their own lines of inquiry as much as possible. In a guided inquiry the teacher might conduct a discussion about the meaning of the regulatory card Change the prompt.
Such a discussion could involve using the What-If-Not strategy from The Art of Problem Posing. What if the integers are not consecutive? What if the integers are not even? What if there are not three integers? What if we do not calculate the sum?
February 2025
Lines of inquiry
- The sum of m consecutive integers
Once students know how to prove the generalisation in the prompt, they can move onto other statements involving consecutive, consecutive even, and consecutive odd integers (see the slides). The table shows the results for the sums of between two to six integers. The sum of four consecutive odd integers, for example, is a multiple of eight.
Students might be encouraged to find the results through deductive reasoning (see the picture below), rather than through the inductive process of identifying a common multiple from different examples.
After completing the table, the inquiry could move into another phase of questioning based on the columns and rows.
If you consider the row for four integers (m = 4), for example, why are the sums of consecutive, consecutive even, and consecutive odd integers multiples of two, four, and eight respectively? Students attempt to explain using the algebraic proofs.
Similarly, considering the final column of the table, why are the sums of an even number of consecutive odd integers a multiple of twice the number of integers but the sums of an odd number are the same as the number of integers?
2. The product
A second line of inquiry that develops out of the What-If-Not strategy is to find the multiples of the products of consecutive, consecutive even, and consecutive odd integers.
3. Other sequences
Other lines of inquiry involve linear sequences. Is the sum of consecutive terms in a linear sequence a multiple?
For example, in the sequence generated from the expression 3n - 2, the sum of two consecutive terms is one less than a multiple of six.
What types of expressions give rise to sums that are multiples regardless of the number of terms?
