happy before proceeding question from Khan Academy (Resource in action: includes teacher support – see UM) Note: Although this has not been covered thoroughly on the A Level in recent years and hence legacy resources are a little limited, there are some older text books that contain good explanations eg Pure Mathematics 2 by Bostock & Chandler on Extension: refer to sufficient and necessary conditions for a point to be an inflection point (link to mathematical terminology and proof) – see page from Wolfram Mathworld meanings. Use A3 paper and give students time to share ideas with other groups. Students can use devices to take a photo of the final versions. Home Learning Starter Main teaching Including key questions, key teaching points, models and resources Notes Including Support and Extension Consolidation/Plenary Including key questions and homework 4 Several questions from legacy resources / text books to practice finding composite functions, including backwards questions eg find functions f(x) and g(x) such that fg(x) = (2x+1)3 or similar. Or questions from Khan Academy LO: Differentiate using the chain rule Introduces considering the derivative of composite functions by thinking about translations – allows students to discover chain rule for themselves. See teaching notes for key questions. (could use this for starter if class already confident with composite functions) Then introduce the chain rule formally, both versions i.e. [fg(x)]’ = f’[g(x)]g’(x) and dy/dx = dy/du x du/dx. (not a free resource – sort cards to cut up and show the process of finding the derivative of four functions) Particular examples to cover: Support Maths centre videos available for students needing extra recap at home Several detailed Khan academy videos Plenary (note – will need to leave out second question as students won’t have covered differentiating ln x yet) Home Learning Integral exercise level 1 / 2 (differentiate according to ability) or practice from legacy resources. Or MyMaths/Y2/Chain Rule (but includes some e and ln) Or MyMaths/Y2/Differentiating
using limits legacy resources) to confirm use of rule. Combining product and chain rules Ask students to work in pairs to differentiate . Emphasise the importance of setting work out very clearly; complete the chain rule part as a side calculation then link in clearly to the main part of the working. Practice: There are lots of legacy resources in textbooks etc on this to practice the techniques, or integral subscribers can use the product rule questions from integral exercise level 1 Extension Finding a repeated root, opportunity for proof (ensure that includes some example where students have to use chain rule in conjunction with the product rule) Or MyMaths/Y2/Product Rule (but includes some e and ln) Starter Main teaching Including key questions, key teaching points, models and resources Notes Including Support and Extension Consolidation/Plenary Including key questions and homework 6 Students to differentiate y = 2x/(x+1) by writing it as 2x(x+1)-1 and using the product rule. LO: Differentiate using the quotient rule (and choosing appropriate rule) Students to establish quotient rule by writing u/v as uv-1 and using the product rule (note, this will require a very careful application of the chain rule in the general case – weaker students will need support with this) Once the quotient rule has been established, students to check their answer to the starter. NB Students may ask why they can’t just always use the product rule by writing a quotient as uv-1. Emphasise that subsequent algebraic manipulation and simplification is often made much easier by using the quotient rule. Types of Differentiation. (not a free resource) Students to sort expressions into a 3 way Venn re how they would differentiate; product vs quotient vs chain and then differentiate as many as they can (will need to leave out differentiating expressions involving e and ln etc for now but can still sort into Venn diagram) Or Support Khan academy , includes derivation of quotient rule from product and chain rules. Pair strong with weak for support Extension Combines with curve sketching and problem solving Plenary “Mark my homework” style problem. Teacher prepared solution to a quotient rule problem containing deliberate errors for students to identify and correct. Could give a problem with harder to spot errors to stronger students. Or The from Kangaroo Maths! Home Learning Questions from legacy resources or integral exercise level 2 Or MyMaths/Y2/Quotient Rule
Product and quotient rule differentiation game from (you will need to take out cards involving exponentials and trig) Or Practice from legacy text books. Starter Main teaching Including key questions, key teaching points, models and resources Notes Including Support and Extension Consolidation/Plenary Including key questions and homework 7 (but will need to change the functions as this involves exp) LO: To use the chain, product and quotient rules in contextual and geometric problems (including connected rates of change) and identify which is the most appropriate. Free resource from Integral Contextual chain rule problems from lesson 4 if not already covered then. Consolidation and practice using legacy text book resources – ensure include a selection of all types. [This section is under construction on undergroundmathematics so very limited selection of free resources other than from legacy text books!] Extension Integral exercise level 3 (product and quotient rules) Plenary Review of entire topic – students to RAG rate different aspects Home Learning Integral Topic Assessment Or MyMaths/Y2/Rates of Change