The prompt is suitable for older students in secondary school and has led to an inquiry that combines mutually-supporting inductive and deductive reasoning. On first seeing the prompt, classes usually interpret the sequence as arithmetic, although it is advisable for the teacher to establish this constraint if it does not arise in discussion. Students' initial questions and comments about the prompt have been:
- Is there one solution, a finite set of solutions, or an infinite number?
- Can we work this out with algebra?
- Is the sequence linear? What kind of sequence is it?
- Is the sequence ascending or descending? Does it matter?
- What if one of the numbers is negative?
- Can they both be negative? Or both be decimals?
- It won't work if they are both negative.
- 1 and 5 work if we change the order: difference 4, product 5, sum 6.
- If one of the numbers is 1, then the sum is greater than the product.
- 2 and 5 do not work: difference 3, sum 7, product 10.
Often the solutions (6, 2) and (4, 4) arise early in the inquiry. Whether they do or not, students often decide to explore or "find more examples" when given a choice of the regulatory cards. The teacher can plot the points on a graph as the class finds more pairs of numbers that satisfy the prompt. This can lead to speculation about how the curve develops, which, in turn, enables students to narrow their search to a range of y values for a specific value of x. For example, when x = 5 the value of y must lie between 2 and 4 given that (6, 2) and (4, 4) satisfy the conditions in the prompt.
(1) A graphical representation
If the two numbers are considered as a coordinate pair (x, y) with x ≥ y, then students can plot a graph. Further inquiry of the asymptotes at x = 3 and y = 1 has been sparked by comparing the algebraic equation and graph together. (A graphing package - such as Desmos - allows students to explore more rational functions.)
(2) An algebraic representation
Students, under the guidance of the teacher or independently, have developed an equation giving the relationship between the two numbers. Once students have y in terms of x or x in terms of y, they go on to substitute values for x or y into their equation to find more pairs of numbers that satisfy the prompt.
(3) Connecting the two numbers to the nth term
One popular line of inquiry in the classroom involves attempting to connect the two numbers to the nth term of the sequence. A table helps students to organise their results and can lead to a recognition of the coefficient of n as 2y. However, the constant is more difficult to identify from the table. As the first term (n = 1) of the sequence is created by x - y, then x - y = 2y + c and, it follows, c = x - 3y.
(4) Further pathways
Even though students have been involved in substantial mathematical thinking up to this point, the prompt has the potential for a richer inquiry. To encourage further questioning, the inquiry teacher could use the strategy laid out in The Art of Problem Posing (see the post on questioning in inquiry):
(Level I) List the attributes
- difference, sum and product in that order
- two numbers
- first three terms
- (arithmetic) sequence
(Level II) Modify the attributes by asking "What-if-not?"
- What if the difference, sum and product were not in that order?
- What if we did not have only the difference, sum and product?
- What if there were not two numbers?
- What if we did not need the first three terms?
- What if the sequence was not arithmetic?
(Of these only the fourth question will not yield an independent inquiry as the number of terms is dependent on the first part of the prompt.)
(Level III) Pose questions
- Would we get an arithmetic sequence if we changed the order of the difference, sum and product?
- Would we get an arithmetic sequence if we used the quotient to give us a fourth term? Where in the order would we place the quotient?
- Is it possible to use more than two numbers to generate the terms?
- Could we create a quadratic or geometric sequence in the same way?
(Level IV) Analyse the questions
The inquiries that now follow tend to involve individuals, pairs or groups choosing to answer one of the questions above with some form of feedback to the rest of the class.
The prompt was inspired by a problem posted on twitter by Steve Leinwand (@steve_leinwand). His problem was as follows: "The sum, difference, product and quotient of two numbers are the first four terms of an arithmetic sequence. What is the value of the fifth term?" Kier Tipple (an Assistant Headteacher in Brighton, UK) has subsequently worked on the problem and his solution can be viewed here.
Structured inquiry sheet This document shows how one teacher structured a classroom inquiry after the initial phases of questioning and regulating inquiry.
In discussion about this prompt, Mike Ollerton (www.mikeollerton.com) has commented that there are an infinite set of solutions if the prompt listed difference, product and sum in that order. For a pair of numbers x and 1, the first three terms of the sequence are x - 1, x, x + 1. He also expressed the difference, sum and product (as listed in the original prompt) as a, a + d and a + 2d to arrive at the result above.