Y12 AS Mathematics Trigonometry 2 weeks Teaching objectives a To review knowledge of trigonometry from GCSE. b To ensure students understand and have efficient routines for finding missing sides and angles in right angled triangles and can apply them in problem solving situations. c To understand, sketch and use the graphs of the sine, cosine and tangent functions as well as transformations of these. d To solve simple trigonometric equations, finding all the solutions in a given range (including quadratic equations in sin, cos tan and equations involving multiples of the unknown angle) d To understand use the identities tanx=sinx/cosx and sin^2x+cos^2x=1, including using them to help solve trigonometric equations. e To ensure students understand and have efficient routines for finding missing sides and angles in right angled and non-right angled triangles. f To understand and know efficient routines for finding the area of a triangle (with or without the perpendicular height). g To gain fluency and develop problem solving skills in the context of trigonometry Notes: Lesson 1 could be done entirely via starters and HW Lessons 3,4,5 could be squeezed into two lessons if time is tight A strong group with good recall from GCSE could do lessons 7 & 8 in one Starter Main teaching Including key questions, key teaching points, models and resources Notes Including Support and Extension Consolidation/Plenary Including key questions and homework 1 Students to produce mind map or similar detailing all their remembered knowledge of trigonometry. Share in pairs/ groups, bring together key points as a class. To be included: formulae, diagrams, graphs. This will be revisited at the end of the unit. SOHCAHTOA problem solving questions. Eg. By drawing an equilateral triangle find the exact values of sin30, cos30 and tan30 and for 60. By drawing a right angled triangle find the exact values of sin45, cos45, tan45 Extension Qs: MEI C4 paper June 2016 Q5 (in part 1 change sec^3(θ)=2 to cos^3(θ)=1/2) Students share solutions Homework: Routine SOHCAHTOA practice from old resources
and resources Extension homework 4 Starter: Quadratic eqns related to trig ones used later– see attachment ‘Quadratic trig equations worksheet’ Match each trig question with a similar quadratic question from starter. Then solve. Equation set 3 from: Last 2 questions in the starter are more challenging due to inclusion of x & y. Stronger students can be encouraged to use correct mathematical notation in their solutions. Plenary or extension Q Homework: Routine practice from old resources/ textbook 5 Mini-whiteboards: Practice solving trig equations Extend to multiple angles. Deliberate error task - card sort. (see attachment – ‘Trig equations with multiple angles error card sort’) Students to identify correct cards and order them, and to describe the mistakes made in the other cards. Tarsia activity: Ask stronger students to generalise- How can you always avoid making mistakes when the subject of the function is more complex than x? Feed back methods to class HW: Routine practice from old resources/ textbook 6 Starter should prompt the need to find out where sinx=cosx, tanx=sinx, tanx=cosx. Discuss ideas of how to do this and bring out need for algebraic solution. Students draw right angled triangle, label and write down SOHCAHTOA rules. Ask them to work out sinx/cosx and sin^2x+cos^2x from those rules to discover identities. Apply to solving above problems. (Equation set 3 may have been used earlier) Pair strong with middle, middle with weak and be prepared to support any of the middle/weak pairings. Ask stronger students for how to apply the trig identities to the problems. For tanx=cox how can we avoid losing the 0 solution? Plenary or extension Q: HW Research proof of trig identities for angles larger than 90 Or fluency practice 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 Students present Proof of sine rule – individual or pair work (you could direct them here to How would you explain why
proofs of trig identities for angles larger than 90 (or later if more time needed for prep) If using 1 lesson, go straight into proof of sine rule task. Recap of cosine rule and area of a triangle. Problem solving questions sourced from old exam papers. Eg from (adjustment may be needed for questions in radians) help with HW) Ability groupings can be used, with all questions needing to be completed by each group, then time given for each student to explain their hardest question to others in their group. If using 2 lessons more time can be spent on sharing ideas and solutions in groups/ as a class. there are two triangles with the properties ? HW Research proof of cosine rule and area of a triangle (especially if 2 lessons used, at beginning of 2nd lesson students can present their proofs) Or fluency practice – including ambiguous case 9 Students to revisit earlier mind maps and add new knowledge. Review lesson with mix of problem solving/ exam style questions. Eg. includes knowledge of circles Some old style exam questions (needs adapting to remove radians): Mini plenaries where stronger students present how they have started harder problem could help support weaker ones. Alternatively pair or group students by ability. Discuss key questions and approaches Homework: Topic assessment