NCETM_Mathematics_Department_Workshop_Sequences

Mathematics Department Workshops Topic: Sequences Overview This unit explores the teaching and learning of sequences. You will be invited to consider generalisation, its relationship to algebra, and what is important when developing learners’ understanding of sequences. You will then explore some strategies and activities which may support you in the planning of a group of lessons to include in your scheme of work focussing on sequences. Objectives related to this topic are: • describe integer sequences; generate terms of a simple sequence, given a rule (e.g. finding a term from the previous term, finding a term given its position in the sequence) (Year 7) • generate sequences from patterns or practical contexts and describe the general term in simple cases (Year 7) • generate terms of a linear sequence using term-to-term and position-to-term rules, on paper and using a spreadsheet or graphics calculator (Year 8) • use linear expressions to describe the nth term of a simple arithmetic sequence, justifying its form by referring to the activity or practical context from which it was generated (Year 8) • generate terms of a sequence using term-to-term and position-to-term rules, on paper and using ICT (Year 9) • generate sequences from practical contexts and write and justify an expression to describe the nth term of an arithmetic sequence (Year 9) • find the next term and the nth term of quadratic sequences and explore their properties; deduce properties of the sequences of triangular and square numbers from spatial patterns (Year 10) Materials required • Resource sheets HT1.SEQ.1 to HT1.SEQ.6 Suggested activities Activity 1: Getting Started Ask the members of your team to look, individually, at the dot patterns on resource sheet HT1.SEQ.1 (If you have the Interactive Mathematics resource Glimpses this can be used as an alternative). Once you have arrived at an answer for each pattern, describe how you ‘counted’ the dots to a partner. Did you all count the dots in the same way? (This activity can also be considered as a sample learning resource for learners). Consider the way that you counted the dots. For each dot pattern assume that this is the third of a growing sequence of dot patterns. Describe or draw the next and the preceding pattern in the sequence. Ask the members of your team to read the extract below from the NCETM Mathemapedia entry on Algebra. What algebra was involved in the counting the dots task? Traditionally algebra has been introduced as ‘manipulating letters as if they were numbers’, which lacks motivation and efficacy and suggests that there are a lot of rules to be learned in order to ‘do algebra’. Another approach is to prompt learners to express generalities as early as possible, and gradually to move to more and more succinct expressions, often in situations in which there are multiple expressions for the same thing. Learners then develop the wish to work out how to transform one expression into another, and they can draw upon their knowledge of arithmetic in doing this. Thus algebra emerges as a language which can be used to express and to manipulate generalisations. www.ncetm.org.uk A Department for Children, Schools and Families initiative to enhance professional development across mathematics teaching

Activity 2: Sample learning activities As a team, use the following resource sheets to explore the suggested learning activities. Slideshow HT1.SEQ.2 – Building sequences using tiles Learners work in pairs to consider the questions asked by the PowerPoint presentation (If you have access to Numicon© tiles then learners could build the sequences rather than sketch them on their whiteboards) Resource Sheets HT1.SEQ.3a, b and c – Reading Images The central image is the first term of each sequence. Learners explain what is happening in each spider diagram and use this explanation to title the diagram (for example, a title may be growing sequences from three dots). They may add to each ‘arm’ of the diagram and/or add an extra arm. Resource Sheet HT1.SEQ.4 – Images of sequences cards Learners match different images of the same sequence. A number of the cards can be placed in more than one category – learners may want to discuss different ways of grouping the cards. Activity 3: Probing Questions Challenge members of your team to write some probing questions using the question stems on resource sheet HT1.SEQ.5. Share ideas and consider how such questioning techniques could be used in your teaching. Some examples of probing questions for sequences can be found here. Activity 4: Reflection – So what’s good about these activities? Read ‘Eight Principles for Effective Teaching’ on resource sheet HT1.SEQ.6 (taken from Malcolm Swan in the Standards Unit ‘Improving Learning in Mathematics’ resource) and the following quote from the Secondary National Strategy’s Teaching Mental Mathematics from Level 5: Algebra booklet Sequences can – deceptively – appear to be straightforward. Consequently the early stages of teaching the topic may be rushed so that pupils do not get the chance to compose the ‘big picture’ for themselves. They need to develop a full understanding of the relationship between the context of a sequence and the ways in which it can be expressed, first in words and later in algebraic terms. It is essential for pupils to understand the term-to-term and position-to-term structures and the relationship between them. Use pupils’ knowledge of variables and functions. T(n) and n are not ‘shorthand’ for words, they are symbols representing variables that can take on specific values (in most examples these are positive integer values). To describe a sequence, we can use either a function that links one term T(n + 1) with the previous term T(n) or a function that links a term T(n) with its position in the sequence n In pairs or small groups discuss the ways in which the sample learning activities fit and don’t fit with these principles. Collect other sequences activities which fit with some of the Eight Principles. Sort these activities into a progression which can be developed into a unit of work to be included into your team’s scheme of work Embedding in practice Hooks for Learning • nrich matchstick problem • nrich coordinate problem • waldomaths sequences Action points At the end of the session, spend time recording some actions. What do you need to do: • Next day? • Next week? • Next year? www.ncetm.org.uk A Department for Children, Schools and Families initiative to enhance professional development across mathematics teaching

Further reading Mathemapedia Algebra and Expressing Generality entries NCETM community discussion Project Mathematics Update: Expressing Generality by John Mason (Open University) Foundations of Algebra Secondary National Strategy www.ncetm.org.uk A Department for Children, Schools and Families initiative to enhance professional development across mathematics teaching