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Extension Y7 AUTUMN TERM UNIT: Algebra 1 - Sequences and Functions TIME ALLOCATION: 6 Hours PRIOR KNOWLEDGE KEY WORDS STARTER Recognise and extend Sequence, term, nth term, 30 starters (subtangent) number sequences, such as consecutive, predict, rule, the sequence of square generate, continue, finite, STARTER OF THE DAY – numbers, or the sequence infinite, ascending, substitution of triangular numbers. descending, symbol, expression, algebra, STARTER OF THE DAY – make Read and plot coordinates substitute, trial and a connection in all four quadrants. improvement, plot, quadratic SUM OF THE SIGNS Understand and use the relationships between the four operations, and principles of the arithmetic laws. Use brackets. LEARNING OBJECTIVES LEARNING OUTCOMES Level 5 To be able to generate and describe a Know that a sequence can have a finite or sequence infinite number of terms The sequence of counting numbers 1, 2, 3, … is To be able to use letter symbols to infinite and the sequence of 2 digit numbers represent unknown numbers and (where both digits are the same) is finite variables To be able to write down first 5 terms and 10th term, given the nth term of sequences such as: 5n + 4 100 - 10n 3n – 0.1 105 – 5n n x 0.1 Use a spreadsheet or graphical calculator to find particular terms such as The 24th multiple of 13 in the sequence The 100th multiple of 27 The nth mutliple of 18 To be able to generate a sequence Use a spreadsheet to generate tables of values when given the position-to-term rule. and explore term-to-term and position-to- term linear relationships To represent mappings expressed algebraically Level 6 To begin to use linear expressions to Find the first few terms of the sequence and describe the nth term of an arithmetic describe how it continues using a term-to- sequence, justifying its form by term rule referring to the activity or practical Describe the general (nth) term and justify context from which it was generated. the generalisation by referring to the context Example – Growing triangles This generates the sequence 3, 6, 9, … Possible explanations ‘We add 3 each time because we add one more dot to each side of the triangle to make the next triangle’ ‘It’s the 3 times table because we get’ To be able to generate sequences from practical contexts The nth term is 3n justification ‘This follows because the 10th term would be 3 lots of 10.’ Develop an expression for the nth term for sequences such as To begin to use linear expressions to 7, 12, 17, 22, … 5n + 2 describe the nth term of an 100, 115, 130, 145, … 15n + 85 arithmetic sequence 2.5, 4.5, 6.5, 8.5, … (4n +1)/2 -12, -7, -2, 3, … 5n – 17 4, -2, -8, -14, … -6n + 10 LEVEL 7 To be able to generate a sequence Find the first few terms in the sequence, using position-to-term definition of describe how it continues using a term-to-term the sequence rule To be able to describe in symbols the rule for the next term or nth term in a sequence (Quadratic) n2 2n2 + 2 n2 – 3 ACTIVITIES ICT RESOURCES Exploring primes activities: MATHSNET algebra topics Numbers of factors; factors Mymaths Square number sequence of square numbers; Mersenne Algebra, sequences Squares in Rectangles primes; LCM sequence; Goldbach's theorem; n² and Nth term generator (n + 1)²; n² and n² + n; n² + 1; Match up n! + 1; n! – 1; Quadratic Generator ~ Venn diagrams for HCF / (with answers) LCM Sequences (slide bars) Taria (Matchup) will need to download software (free) KS3 Y8 Intervention ~ Lesson 8N1.1 Solving number problems 2 FUNCTIONAL SKILLS and MPA OPPORTUNITIES Cuisenaire Rods – Interactive Using only 2 rods make all cuisenaire rods Cross-curricular links with music – sequences generated by beats and rhythm Rich Learning Task: Swimming Pool PLENARIES AND KEY QUESTIONS What did you look for in your sequence to help you find the nth term? How does this link to ... ? (Use the context that generated the sequence.) Probe further to get pupils to justify specific parts of the generalisation – e.g. explain why 'multiply by 4' is part of your nth term. The term-to-term rule for a sequence is 'previous term + 2'. What does that tell you about the position-to-term rule? Do you have enough information to find the rule for the nth term? Why? What do you look for in a sequence to help you to find the position-to-term (nth term) rule? How would you go about finding the position-to-term (nth term) rule for this information on a sequence: Position 3 5 10 Term 11 19 39 Compare a linear to a quadratic sequence. What do you notice about the differences between succeeding terms? What clues do you look for when deciding whether a sequence is quadratic? What can you say about the nth term for a quadratic sequence? What strategies do you use to find the nth term for a quadratic sequence?