It Stands to Reason
For many students, one of their early encounters with using letters or symbols to represent numbers in secondary school is when finding an expression for the value of the nth term of a linear sequence. Although students will have had experience of algebraic representation in their primary schools, the associated task will, most likely, have been to find the value of the unknown symbol; the use of a symbol to mark a potential, or future, evaluation will be much less familiar.
The first part of the challenge of teaching this topic is to find out what they already know, and then build on that understanding to develop the reasoning skills that are the hallmark of good mathematicians. The NCETM has developed activities and resources to help. In the NCETM Departmental Workshop on sequences there is a PowerPoint presentation that encourages students to develop their understanding of the structure of a sequence and relate that structure to the growth of the sequence:
Seeing this, students may say
“In the first picture there is 1 orange piece and 1 blue piece”
“In the second picture there is 1 orange piece and 2 blue pieces”
“In the third picture there is 1 orange piece and 3 blue pieces”
Asking “what’s the same and what’s different” about the blue and orange pieces should elicit realisation that the blues are “double”. At this point they may want to count the holes not the pieces; if counting a hole – a void – is found to be hard, you can ask “how many grains of rice would I need if I wanted to put one in each hole?”
Then your students can be encouraged to rephrase their original answers in terms of the numbers of “orange holes” and “blue holes”, and then predict that
“In the fourth picture there will be 1 orange hole and … blue holes”
“In the tenth picture there will be …”
“In the hundredth picture there will be …”
They then can be asked “How could you explain how many coloured holes there’ll be in any pattern in the sequence?”
The use of colour is helpful because students can talk about the different parts of the sequence: the constant and the changing parts.
This slide allows students to make comparisons between three different sequences that all grow in the same way (or, and better, “at the same rate”) but have different constant terms. Pupils develop their understanding of the structure of sequences, and use the observed structure to predict future specific values. The guiding question “How could you explain how many holes there’ll be in any pattern in the sequence?” provides a link to talking about an expression for the value of the general term of the sequences of the number of holes in each pattern.
A natural follow-on from this resource is to give students an opportunity to create their own linear sequences using a practical resource that enables the use of two or more colours. The blog entry nth term from matchstick patterns gives some more suggestions for relating the structure of a sequence to its growth, and observes that “John Mason suggests providing just e.g. the third diagram in a growing sequence. Following class agreement on the 'structure' of the growth, a generalisation can then be sought. The shared development, seeking alternatives, and subsequent agreement on a 'structure' he maintains is an important part of such work.”
Teaching Mental Mathematics from Level 5 – Algebra (Page 13) gives further useful prompts to ask students to relate the observed structure of a linear sequence to the way that, and the rate at which, it grows, and from there to writing an expression for the value of the nth term.
The activity Shifting Times Tables from NRICH gives a different perspective on finding an expression for the value of the nth term of a linear sequence, by considering the link between the numbers in a sequence and the related times table:
The activity displays the terms of a linear sequence and asks the students to identify the related times table and then the shift that has been made to the times table to get the numbers in the sequence; from there an expression can be given for the value of the nth number in the shifted times table. The different levels in the activity increase in complexity: levels 1 & 2 always display consecutive terms of a linear sequence whereas levels 3 & 4 display any terms of a linear sequence.
There are bound to be different answers – not least of the “5n + 2” vs. “5n – 3” variety – and so there will be plenty of opportunity for students to present to each other and to discuss each other’s suggestions. Peer challenge – and robust defence! – is a very powerful tool for developing reasoning skills, and gives many opportunities for developing the oral skills that the new Programmes of Study quite rightly highlight.
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