Forming a sequence inquiry
Mathematical inquiry processes: Explore; generate examples and counter-examples; find sets that satisfy conditions. Conceptual field of inquiry: Position-to-term rules; algebraic expressions; graphing a solution set.
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.
Phases of inquiry
These are the questions and observations about the prompt from a year 10 class with high prior attainment in mathematics lessons.
By the end of the first 100-minute lesson, students had developed three distinct representations: numerical, graphical and algebraic. At the end of the lesson, the approaches taken by the class were summarised on the board (see illustration) with the teacher noting that the two formulae giving x in terms of y were different.
The second phase of the inquiry (lasting for a 50-minute lesson) was directed by the teacher to ensure all students understood the mathematical content of the first phase. It started with the teacher clarifying that x > y for the purposes of the inquiry. Students who had induced x = y/(y-3) from numerical examples had inadvertently assumed y > x. Some students then set out to show that y = x/(x-3) is equivalent to x = 3y/(y-1) by substituting in the value of y in terms of x from the first formula into the second one. Other students substituted negative values of y into the formula and drew the graph of x = 3y/(y-1) with asymptotes at y = 1 and x = 3.
The third (and final) phase, which lasted 100 minutes, saw students decide on which pathway they would like to develop from the initial list of questions and observations. Most of the class continued to select the card 'work as a table'. The inquiries involved (1) changing the order of difference, sum and product; (2) attempting to connect the nth term of the sequence to the pair of numbers; and (3) attempting to generate quadratic sequences (using the quotient as well).
1. 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.
2. 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.
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.
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
first three terms
(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?
(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.
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.
These are the questions from Caitriona Martin's year 10 (higher attaining) class on first seeing the prompt:
Is it in that order - difference, sum and product?
What is the difference in meaning between 'sum' and 'product'?
What relationship does the nth term have to the two chosen numbers?
Do the numbers have to be integers?
What are the next three terms and what is their relationship to the two numbers?
Is the sequence linear, quadratic or cubic?
If you choose two numbers (a, b) to make a sequence, would two
multiples of a and b make a sequence with terms that are in the original one?
One notable result came out of the sequence that was generated from 5 and 10. Using the first three terms (5, 15, 50), one student generated the fourth term by using a term-to-term rule. The teacher then helped to express the rule formally as a recurrence relationship.
Excerpts from students' inquiry reports
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.