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Intersecting sequences inquiry

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.
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 (below). Another pathway involves inquiring into combinations of even and odd numbers in the first two expressions:
  • a and c have a common factor (b, d co-prime)
  • b and d have a common factor (a, c co-prime)
  • a, c common factor; b, d common factor.
  • a, b, c and d are all odd.
  • a, b, c and d are all even.
  • a, b odd; c, d even.
  • a, c even; b, d odd.
  • a, c odd; b, d even.
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.
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 maths and can be contacted through his websiteHe is on twitter @MichaelOllerton.
Algebraic proof
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 maths 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.

Odds and evens sheets Position-to-term rules and term-to-term rules. Initially, students have distinguished between the pairs that can and cannot produce an intersecting sequence.
Guided poster for reporting on the findings of the inquiry.

The Inquiry Maths session at the 2015 conference of the Association of Teachers of Mathematics was based on the intersecting sequences prompt. You can read a report of the session here

Alternative prompts 
Shawki Dayekh, a teacher of mathematics in London (UK), devised these prompts with his year 7 mixed attainment class during the intersecting sequences inquiry in January 2020:
(1) If an + b is an expression for the nth term of a linear sequence with odd numbers only, a is always even and b is always odd.
(2) When an + b and bn + a are expressions for the nth term of 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.
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 box below).

A new inquiry pathway
In February 2017, 25 maths teachers 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, participants noticed that "every second term in the blue sequence matches every third term in the green sequence." Another pair 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, participants created the following results in a short period of inquiry:
Number of terms between 7 and 13 (t)Sequence 
(with two terms before 7)
 Expression for nth term
1 1  4  7  10  13 3n - 2
23  5  7  9  11  13  2n + 1
34  5.5  7  8.5  10  11.5  13 1.5n + 2.5
44.6  5.8  7  8.2  9.4  10.6  11.8  13 1.2n + 3.4
5 5  6  7  8  9  10  11  12  13  n + 4
tGap between terms
(13 - 7) ÷ (
t + 1)  
Coefficient of n  ÷ (t + 1)
The three teachers involved in generating the findings then presented them to the rest of the group. They were Paul Besgrove (Brittons Academy), Jacqueline Mcleod (Bower Park Academy) and Daniel Allen (Hall Mead Academy). 

Jacqueline and Daniel's inquiry notes.
More inquiry pathways
Matt Carvel (a secondary school maths teacher) 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 a choice of two pathways to follow. Both pathways came out of the exploration phase.
Pathway 1
The first pathway originated in an observation made by Olivia and Ezme. If you generate two sequences from expressions in the form a
n – 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
(above). They had found one exception to their rule. When b is a factor of a, the rule does not work (left). 
Students in the class found the rules for similar cases (using co-prime values for a and b):
First two expressions
Third expression for the nth term of the intersecting sequence
an – b
bn – a
abn – (a + b)
an + b
bn – a
abn + (b – a)
an + b
bn + a
abn – ?
Pathway 2

The second pathway came out of Tom’s inquiry into what would happen if a and b were both even. He reported that two expressions of the form E(n) + E would give an intersecting sequence with an nth term also in the form E(n) + E. The students involved in this pathway reported 
the following results at the end of the inquiry:
First two expressions
Third expression for the nth term of the intersecting sequence
E(n) + E
O(n) + O
E(n) + E
O(n) + E
O(n) + E
O(n) + O
E(n) + E
O(n) + E
E(n) + E
E(n) + O
E(n) + O
E(n) + O
O(n) + E
E(n) + O
E(n) + O
O(n) + O
O(n) + E
O(n) + E
E(n) + E
E(n) + O
No common terms in the two sequences.

Matt Carvel is a newly qualified teacher who has used Inquiry Maths prompts since beginning to train. In February 2016, Matt ran the Inquiry Maths session at the Sussex Maths Conference (held at the University of Brighton, UK), using the intersecting sequences prompt to model 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.
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.

You can follow Mary Pardoe on twitter @PardoeMary.
"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 worksheet available below.) 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.

Exploring the sequences in the prompt
A year 8 class (higher attaining students) 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.

This inquiry was carried out by Caitriona Martin's year 8 class. You can follow Caitriona on twitter @MrsMartinMaths.