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YEAR 9: AUTUMN TERM Teaching objectives for the oral and mental activities a) Order, add, subtract, multiply and divide integers. b) Multiply and divide decimals by 10, 100, 1000, 0.1 and 0.01. c) Count on and back in steps of 0.4, 0.75, 3/4… d) Round numbers, including to one or two decimal places. e) Know and use squares, cubes, roots and index notation. f) Know or derive quickly prime numbers less than 30 and factor pairs for a given number. g) Convert between fractions, decimals and percentages. Know that 0.005 is half of one per cent. h) Find fractions and percentages of quantities. i) Know or derive complements of 0.1, 1, 10, 50, 100, 1000. j) Add and subtract several small numbers or several multiples of 10, e.g. 250 + 120 – 190. k) Use jottings to support addition and subtraction of whole numbers and decimals. l) Use knowledge of place value to multiply and divide, e.g. 432 0.01, 37 0.01, 0.04 8, 0.03 5, 13 1.4. m) Recall multiplication and division facts to 10 10. Derive products and quotients of multiples of 10, 100, 1000. n) Use factors to multiply and divide mentally, e.g. 22 0.02, 420 15. o) Multiply and divide a two-digit number by a one-digit number. p) Use approximations to estimate the answers to calculations, e.g. 39 2.8. q) Solve equations, e.g. n(n – 1) = 56, + = –46. r) Visualise, describe and sketch 2-D shapes. s) Recall and use formulae for the perimeter of a rectangle, and areas of rectangles and triangles. t) Calculate volumes of cuboids. u) Estimate and order acute, obtuse and reflex angles. v) Use metric units (length, mass, capacity) and units of time for calculations. w) Use metric units for estimation (length, mass, capacity). x) Convert between metric units, including area, volume and capacity measures. y) Discuss and interpret graphs. z) Calculate a mean using an assumed mean. aa) Apply mental skills to solve simple problems. YEAR 9: AUTUMN TERM Algebra 1 & 2 (6 hours) Sequences, functions and graphs (148–163, 172–177) Solving problems(26–27) Teaching objectives for the main activities CORE From the Y9 teaching programme A. Generate terms of a sequence using term-to-term and position-to-term definitions of the sequence, on paper and using ICT. B. Generate sequences from practical contexts and write an expression to describe the nth term of an arithmetic sequence. C. Find the inverse of a linear function. D. Construct functions arising from real-life problems and plot their corresponding graphs. E. Represent problems and synthesise information in algebraic, geometric or graphical form; move from one form to another to gain a different perspective on the problem. Unit: Algebra 1 & 2 Year Group: Y9 Number of 1 Hour Lessons: 6 Class/Set: Core Oral and Mental Objective b & c and A Counting on and back activities in steps of 0.4, 0.25, ¼, ¾ etc. Main Teaching Lesson 1 – (Could be 2 lessons) Objective A and B Revisit or introduce ‘T’ notation for sequences. Choose different starting numbers. T(1) = first term T(2) = 2nd term etc. Establish these are sequences and Model T(n) = nth term = 2n remind about need for a starting value Model other functions. and a rule. (See page 149 of framework) Extend to multiplying by 10, 100, 1000 etc., By 0.1, 0.2, by 0.01 etc. Set question like if T(n) = 2n – 7 what is T(6)? Establish than adding on and taking off is an arithmetic sequence. Revisit finding rules from input and output tables. Establish difference between term to term and nth term rules. (See www.eril.net following links to learning worlds – maths – interactive whiteboard – sequences and rules) Distinguish between these and geometric sequences. Notes Key Vocabulary Sequence Arithmetic Linear Term to term nth term Difference Plenary Give me an example of an arithmetic sequence Give me a non-arithmetic sequence What is a term to term rule – give an example. How else can we describe a sequence? What does T(n) = 2n – 6 mean? 6, 10, 14, 18….. T(n) = ? How do we find the rule? Extension – what about this sequence 1, 4, 9, 16? How do we describe it? Oral and Mental Objective B (Writing expressions) My starting number is always n. Give me an expression for: Twice n n multiplied by itself Half n The reciprocal of n 25% of n 8 more than n n doubled and six added Six added to n and then doubled Etc. (Pupils respond with white boards etc.) Pupils sketch graphs of functions y = x, y = 2x, y = 2x + 1 etc. in pairs on whiteboards. Main Teaching (2 lessons) Objective D and E Revisit/model function like y = 2x, y = 2x + 2 etc relate x and y values to n 2n, n 2n + 2 etc. show links. Revisit sequence 1, 4, 9, 16, model graph – establish why it is non-linear. Model everyday life situation. e.g. a cement mixer cost £20 to hire plus £5 for each day. Produce Table of Values for up to one week. Day Cost 1 25 2 30 3 35 4 40 Model term to term rules Model nth term rule. Plot graph, make links. Questions needed on practical situations using real life problems. Pupils undertake surround activity (worksheet Surround 9Alg 1&2a ) Notes Plenary Revisit cement mixer activity Compare Table Graph Algebraic rule (T(n) = 5n + 20) and worded rule “£20 + £5 per day” How does each present the situation? Model other questions. Oral and Mental Objective y What type of rules might give these graphs: f(n) n f(n) n f(n) n Main Teaching 2 lessons Objective C Pupils list sequence generated by T(n) = 10 – n e.g. 9, 8, 7, 6 … If I know the number in a sequence, how can I tell which term it is? Establish 10 – sequence number = term number. What about T(n) = 15 – n T(n) = 13 – n What do you notice? Note that ‘undoing’ the sequence we use the same function. Introduce the idea of ‘inverse’ of a function. Model in words a function and then the inverse. Relate to number machines and mappings. e.g. ‘I take a number, double it and then add one’. Inverse is ‘ I take one and halve’. Model the algebra. n 2n + 1 confusion may occur n n - 1 using the n notation 2 f(n) To simplify this you could use y = 2x + 1 with and inverse x=y - 1 2 n Graded questions needed – some students may remain at the function/number machine stage. f(n) n Pupils discuss each on in pairs. Teacher collects responses – class discuss. Notes Key Vocabulary Inverse Self inverse Plenary Model graphs of: T(n) = 10 – n i.e. y = 10 – x T(n) = 13 – n y = 13 – x etc. What do you notice? Talk about self inverses. Plot y = x …… is this a self inverse? What do you notice: OR Carefully remodel functions and inverses. How do inverse functions help? Oral and Mental Objective C ‘I think of a number’ e.g. I think of a number double it and add 6. The answer is 10. What was the number? What was the function? What was the inverse function? Grade questions carefully and support/model responses. Main Teaching Objective A, B, C, D, E (Model the following with help from students) A taxi driver charges £2 as soon as you sit in his cab and the £1 per mile. Write out a table of values for each increasing mile. What is the term to term rule? What is the nth term rule? Write this as a function. Plot the graph. Use all the above to answer the following: How much to travel 12 miles? If I was charged £28, how many miles did I travel? Further questions of this type for pupils to solve. Notes A function’s graph line reflected in the line y = x will give the inverse function. Plenary Discuss solutions Compare graphs of function and inverse Function example with graph line y = x Conjecture? Is this always true? Check with further functions.