Intersecting sequences inquiry
Mathematical inquiry processes: Notice connections; generate examples and counter-examples; conjecture and generalise. Conceptual field of inquiry: Linear sequences; term-to-term and position-to-term rules; algebraic expressions.
On first viewing, the prompt seems closed. Two expressions give linear sequences from which the common terms are combined to make a third sequence. The expression for the nth term of the new sequence is then deduced. However, this inquiry has generated a variety of questions about sequences. It has also led to highly focussed classes who become engrossed in trying to produce a general rule that connects the first two expressions to the third one.
The prompt suggests that for two expressions of the form an + b and cn + d, the expression for the nth term of the intersecting sequence is (ac)n + d. This attempt to generalise turns out to be premature, as a counter-example shows. 2n + 1 and 4n – 3, for example, give an intersecting sequence expressed by 4n + 1. More intriguingly, perhaps, are 2n + 1 and 6n – 1, which give 6n – 1. The example of 4n – 3 and 6n – 1 is also instructive because they lead to 12n – 7. Ultimately, students realise that the coefficient of n in the intersecting sequence is the lowest common multiple of the coefficients of n in the original sequences.
The prompt teaches students an important lesson in their search for a 'rule' about the constant: there is not always a neat solution, or at least one that can be arrived at through spotting patterns. Indeed, at times, each new case seems to contradict a generalisation from the previous ones. Students can begin to appreciate that explaining the inner workings of one case is often better than producing more examples.
Lines of inquiry
Classes have taken this prompt in different directions. At the start, students often aim to reproduce the procedure in the prompt with their own examples, developing fluency in finding an expression for the nth term as they do so. After that, different pathways have developed, including a group of year 8 students who looked at intersecting quadratic sequences using a spreadsheet.
Another pathway involves inquiring into combinations of even and odd numbers as the coefficients and constants of the first two expressions. For example, if the two expressions are an + b and cn + d, then a and b could be odd and c and d even (see 'More inquiry pathways' below).
During the 2021 UK lockdown, Andrew Blair used the intersecting sequences prompt to run an online inquiry with his year 7 mixed attainment class. He reports on the course of the inquiry:
We had carried out one online inquiry before (into the number line prompt). Compared to our face-to-face inquiries in the classroom, I had to structure that inquiry far more and, consequently, it was less responsive to students' emerging questions (see the report here). Even though I had planned to involve students more through the regulatory cards this time, the second inquiry followed a similar course to the first one.
The inquiry started with a tremendous burst of energy as students posed questions about the prompt using sticky notes on a Jamboard (picture above). They noticed the common terms in the three sequences, tried to find a connection between the sequences and started to speculate about one of the algebraic expressions as the "formula of the sequence".
Importantly for the construction of an understanding of position-to-term rules, one student linked the sequences to times tables. I was able to use her idea to describe a linear sequence as a shift of a times tables.
We then used the cards to propose ways forward, but it proved difficult to keep track of the suggestions. In the classroom I might suggest that students with similar ideas or identified weaknesses could work together in mutually supporting groups, but we had not yet developed the levels of independence and initiative to do this successfully online.
In the end, I structured the inquiry through the slides that we had become accustomed to using through Google Classroom. There were two lines of inquiry based on term-to-term rules or position-to-term rules.
Conjectures and proof
Conjectures about the nth term
Students often make the conjecture that the coefficient of n for the intersecting sequence is the product of the coefficients in the two position-to-term rules. On finding counter-examples, they change the conjecture to include dividing by the highest common factor of the coefficients. However, conjectures about the constant will take many forms depending on their examples. You can read three conjectures from year 7 students here.
Chinese remainder theorem
In his book Getting the Buggers to Add Up, Mike Ollerton writes about an intersecting sequences investigation that starts with students (numbered 1 to n) standing in a circle. He contacted Inquiry Maths to suggest that the prompt is related to the Chinese Remainder Theorem. A procedure can be derived from the theorem to find an expression for the nth term of an intersecting sequence. The procedure uses modular arithmetic to produce an equation that can then be adapted to create the nth term. These notes give two examples.
Mike Ollerton writes widely about ideas for teaching mathematics and can be contacted through his website.
Prompts from conjectures
Shawki Dayekh, a teacher of mathematics in London (UK), devised these prompts from conjectures that his year 7 mixed attainment class made during the intersecting sequences inquiry in January 2020:
(1) If an + b is an expression for the nth term of a linear sequence that contains only odd numbers, a is even and b is odd.
This conjecture is true. If a is odd and b is even, then the terms in the sequence alternate between odd and even.
(2) When an + b and bn + a are expressions for the nth term of the linear sequences P and Q respectively, then the second term of sequence P equals the third term of sequence Q if and only if a = 2b.
Testing the conjecture with a = 4 and b = 2 gives 6, 10, 14 for sequence P and 6, 8, 10 for sequence Q. In general, if we substitute 2b for a, then 2b(2) + b = b(3) + 2b.
George Marsden (a year 10 student at St. Andrew's School, Leatherhead, UK) ended his inquiry by proving the following statement: the product of any two terms in the sequence given by the expression for the nth term 6n + 1 is also a term in the same sequence.
This is an impressive observation and proof. Helen Hindle, a secondary school mathematics teacher, has subsequently generalised the observation to all cases in creating the prompt The product of any two terms in a sequence is also a term in the sequence.
Questioning and noticing
These are the questions and observations of a year 10 class at Haverstock School in Camden (London, UK). The students have noticed a connection between the coefficients of n in the position-to-term rules. One pair have speculated that there is no link between the 'statics' (or constants). Another inquiry pathway is opened by the observation about the gaps between 7 and 13 in the sequences. This triggers the questions about whether sequences can be generated with increasingly more gaps and if there is a pattern or connection between the expressions for the nth terms of those sequences (see 'A new line of inquiry' below).
Exploring the sequences in the prompt
Caitriona Martin's year 8 class decided to inquire into the sequences in the prompt. In the illustration above, students have identified the multiples of five (given by 30n - 5). They have also started to consider the pattern of square numbers, conjecturing that only the squares of prime numbers appear in the sequence. As 17 is the next prime number after 13, they predict that the next square will be 289.
While the prediction turns out to be true, the reasoning is flawed. The first square of a composite number to appear in the sequence is 625. The list of the numbers whose squares appear in the sequence (5, 7, 11, 13, 17, 19, 23, 25, 29, 31 and so on) shows that the squares of the odd multiples of three are not in the sequence. This is because if a number is a multiple of three, its square will also be a multiple of three. Any term in the sequence generated by 6n + 1, however, cannot be a multiple of three. In a second inquiry (shown below), a student focussed on the terms that were in neither of the sequences produced from the expressions 3n - 2 and 2n + 1. The result was an 'anti-sequence', as the class called it, that was shown to be linked to the sequence 6n + 1.
"I notice that ..."
On being asked to finish the sentence "I notice that ...", a year 10 foundation GCSE class came up with the board above. Students then selected cards, from which the teacher designed the following inquiry sequence:
The teacher explains.
Students look for more examples.
The class shares its results.
The results are discussed as a class.
At the end of the first lesson, students had selected their own pairs of expressions for nth terms, produced the sequences and started to identify types of pairs that do not have common terms. The second lesson started with the teacher explaining how to find the nth term of a sequence. Students deduced the nth terms of the intersecting sequences they had created in the first lesson and realised that the coefficients of n are linked. Some students opted to use the sheet available below in 'Resources'.
In the final lesson, the class used the observations from the first lesson to explore which pairs of nth terms have no common terms by considering the role of odd and even numbers.
New lines of inquiry
Matt Carvel (a secondary school teacher of mathematics) used the intersecting sequences prompt with a year 9 class. In the first lesson, the students posed questions and made comments before exploring the prompt. At the start of the second lesson, Matt guided the class towards following one of two lines of inquiry that had came out of the exploration phase.
Line of inquiry 1: an - b and bn + a
The first line of inquiry originated in an observation made by Olivia and Ezme. If you generate two sequences from expressions in the form an – b and bn + a, then the expression for the nth term of the intersecting sequence will be abn + (a – b). They wrote their rule for one case on the board (see below).
They had found one exception to their rule. When b is a factor of a, the rule does not work. For example, 4n - 2 and 2n + 4 do not fit the generalisation. Students in the class found the rules for similar cases using co-prime values for a and b.
Line of inquiry 2: Odds and evens
The second line of inquiry came out of Tom’s inquiry into what would happen if a and b were both even. He reported that two expressions in the form E(n) + E would give an intersecting sequence with an nth term also in the form E(n) + E.
Matt suggested the students try other combinations. What would happen, for example, if both coefficients and constants are odd? Or both coefficients are odd and both constants are even?
The students reported on their findings at the end of the inquiry. Matt recorded the results in a table.
The last case led to a discussion about why there were no common terms. For the first expression, when n is odd or even, the terms are even. The students represented this as E x O + E = E and E x E + E = E. Similarly for the second expression, E x O + O = O and E x E + O = O.
In February 2016, Matt Carvel ran the Inquiry Maths workshop at the Sussex Maths Conference (held at the University of Brighton, UK), using the intersecting sequences prompt to demonstrate how inquiry develops in the classroom. In the picture, Matt is handing out the regulatory cards for the 25 participants to decide how their inquiry might develop.
Line of inquiry 3: Number of terms between 7 and 13
In February 2017, 25 teachers of mathematics from the three schools within the Empower Learning Academy Trust (London Borough of Havering) participated in an Inquiry Maths workshop. The session saw the emergence of a new approach to the intersecting sequences prompt.
In the first phase of the inquiry, a pair of teachers noticed that there is one term between 7 and 13 in the first sequence (3n - 2) and two between 7 and 13 in the second one (2n + 1). They wondered if there is a relationship between the first two sequences in the prompt and one that has three terms between 7 and 13.
Selecting the regulatory cards related to extending relationships and finding connections, the pair (who were joined by another teacher) created the results shown in the table. The three then presented their findings to the rest of the workshop participants.
The three teachers were Paul Besgrove (Brittons Academy), Jacqueline Mcleod (Bower Park Academy) and Daniel Allen (Hall Mead Academy). The illustration shows a page of their notes.
In May 2022, participants in an Inquiry Maths workshop at Ecolint, an international school in Geneva, developed two new lines of inquiry.
Line of inquiry 4: Formula
Three teachers devised a formula to find the expression for the nth term of the intersecting sequence, which they presented to the other participants (see the illustration above). The formula is:
(f - e)n + (e - (f - e))
where e is the first common term of the intersecting sequence and f is the second.
For the case in the prompt, e = 7, f = 13 and the nth term is (13 - 7)n + (7 - (13 - 7)) = 6n + 1
Example 1: 5n - 2 and 2n + 3
The first two common terms are 13 and 23. The expression for the nth term of the intersecting sequence is given by (23 - 13)n + (13 - (23 - 13)) = 10n + 3.
Example 2: 3n - 4 and 4n - 3
The first two common terms are 5 and 17. The expression for the nth term of the intersecting sequence is given by (17 - 5)n + (5 - (17 - 5)) = 12n - 7.
Line of inquiry 5: Composite functions
Emilie Montanes and Jayne Bryant, who were training on the international PGCE course at Durham University, used the concept of composite functions to launch a line of inquiry.
They noticed that 3(2n + 1) - 2 = 6n +1. This is an intriguing result, which suggests that f(g(x)) is the nth term of the intersecting sequence. However, g(f(x)) does not work because 2(3n - 2) + 1 = 6n - 3.
Emilie and Jayne found other examples:
Example 1: 4n - 3 and 3n + 2
The intersecting sequence is 5, 17, 29, ...
3(4n - 3) + 2 = 12n - 7, but not 4(3n + 2) - 3
Example 2: 5n - 4 and 2n + 1
The intersecting sequence is 11, 21, 31, ...
5(2n + 1) - 4 = 10n + 1, but not 2(5n - 4) + 1
in general, for an + b and cn + d, a and b must be co-prime. Either ad + b or bc + d equals the constant in the expression for the nth term.
The composite functions approach worked for only a few of the intersecting sequences that Emilie and Jayne tried. They did not find an example for which f(g(x)) = g(f(x)) = the expression for the nth term.
An alternative representation
The prompt generated these questions and comments from a year 7 mixed ability class who had carried out four mathematical inquiries previous to this one. Students speculate on the missing numbers, notice patterns related to the types of numbers in the sequences, and begin to link the expression for the nth term to its corresponding sequence. Indeed, one pair of students is confident enough to suggest changing the expressions, and another moves towards linking the first two expressions with the third.
When these responses were posted on twitter, Mary Pardoe suggested that visual images could help students see the connection between the two sequences and their intersecting sequence. In the image (below), the sequences generated by 2n + 1 and 3n - 2 are shown on the left and right respectively. The common terms are highlighted in bold and split into blocks of six squares with the one at the front to illustrate 6n + 1. This is a highly attractive representation of the prompt and represents another potential pathway for the inquiry. However, as discussed in the y - x = 4 inquiry, students new to inquiry rarely suggest an alternative mathematical representation spontaneously. The teacher will need to introduce the different representation explicitly and explain how it can enrich the students' understanding of the underlying structure of the prompt.
ATM conference 2015
The ATM conference 2015 was held in Daventry (UK) with the theme "Thinking Mathematically".
The Inquiry Maths session at the Association of Teachers of Mathematics conference 2015 attracted 40 participants. We looked at the intersecting sequences prompt, coming up with a wide variety of questions and observations:
What other sequences have 7 and 13 in? Is it a coincidence that 7 and 13 are prime?
The sequence generated by n + 6 also contains 7 and 13.
The sequence generated by 1.5n - 2 also contains 7 and 13.
Are there other numbers that are in both sequences?
How does the red sequence lead to the red expression?
What is the link between the two expressions and the last one?
The product of the coefficients of n in 3n - 2 and 2n + 1 give the 6, but I don’t know how to find the constant.
The differences between each term in the first two sequences are -2 -1 0 1 2 and the expression for the nth term of that sequence is n - 3.
If you sum the three expressions 3n - 2, 2n + 1, and n - 3, you get 6n - 4. The coefficient is correct, but not the constant.
Looking at the numbers between 7 and 13, there are none in the red sequence, one in the blue, and two in the green. What would the expression be for the sequence with three numbers between 7 and 13?
If you arrange the expressions for the nth terms vertically, what expressions would come before and after the ones in the prompt?
We notice that for odd and even numbers: 3n - 2 gives O E O E; 2n + 1 gives O O O O; and 6n + 1 gives O O O O.
Three participants presented their work in progress:
Josh Evans (a teacher from Steyning Grammar School, UK) showed us how the lowest common multiple could be used to find the coefficient of n. 3n - 2 and 5n + 3 lead to 15n - 2 and the lcm(3,5) = 15. Similarly 2n + 1 and 4n + 3 give 4n + 1 and the lcm(2,4) = 4. He speculated that the constant comes from one of the first two expressions (-2 and +1 in his examples). A counter-example was produced by another participant.
The second contributor showed how any expression with a coefficient for n of 1, 2, 3, or 6 would give a sequence including 7 and 13. This can be achieved by ‘sliding’ the multiplication table until it gives 7 and 13. For example, 2n gives 2 4 6 8 10 12 14. You could slide this 5 to the right (2n + 5) or 3 to the left (2n - 3) and so on.
Finally, Anna Dickson (a teacher who now works for the Oxford University Press) showed how she had started to use modular arithmetic to generate expressions for sequences that start with 7. She demonstrated her work using modular “clocks”:
(6/1)n + 1 where 1 = 1 (mod 6)
(6/2)n + 4 where 4 = 1 (mod 3)
(6/3)n + 5 where 5 = 1 (mod 2)
(6/4)n + 5.5 where 5.5 = 1 (mod 1.5)
This generalises to mn + 1 (mod m) where 7 - m = 1 (mod m). Therefore, we have a general expression to create a sequence that starts with 7: mn + (7 - m).