Web viewStarter puzzle with differences of two squares. ... getting students to demonstrate answers...

A101

A101a ALGEBRA INDICES SURDS2.1 Understand and use the laws of indices for all rational exponents2.2 Use and manipulate surds, including rationalising the denominator

4 hours guided learning time

BAT manipulate algebra with some fluency; simplify algebra fractions using factorization Key Q: how do we rearrange equations using inverse operations; when is factorisation useful for simplifying;

BAT use substitution to find values and transform functions;Key Q: how can we substitute correctly to expressions; what is the correct order of operations;

A1011 WB1-4

Starter is rearranging equations RAG. Use for AFL to diagnose any misconceptions.WB1 is two more rearranging equations with manipulation in several steps including refactorizing to get single term of x. WB2 simplify algebra fractions and simplify by cancelling. Four examples. Possibly stop here and do practice worksheets. WB 3 is substituting values to formulas followed by peer coaching activity choosing a function and substituting working in pairs; re-group students appropriately. Chance to explain to students the benefits of working together and checking strategies in lessons. Basic function notation f (x) being used and can be discussedWB4 is substituting expressions to formulas followed by similar peer coaching activity. Extend by getting students to make up more questions and try on each other.

BAT apply the laws of indices to calculations; should know equivalence of n√ xm=xm /n ;Key Q: which rules of powers are tricky and when;

A1012aA1012bA1012cWB 5 to 6

Starter indices bingo - Use as AFL and ask students which rules of indices they are weak at using. Get students working in pairs and competing. WB5 set of exemplar questions for students to use as a reference. In particular model finding harder fraction powers by finding fraction part first – breaking into steps. Then practice using worksheets RAG. Make sure students are practicing skills appropriate to need. And that work set for independent study to bring skills up to fluency with fraction and negative indices. Followed by WB 6 harder questions to check understanding. Plenary: indices countdown. Identify any types of Qs students still struggling with

BAT manipulate expressions using rules of indices; use function notationKey Q: how do the rules of powers work with algebra expressions and

A1013aA1013b WB 7-

Starter numerical calculations with powers including fraction powers RAG. WB7-8 is a set of exemplar powers of algebra such as reciprocals, brackets to go through and for students to have as a reference. Once students happy with these:

fractions – what do we need to be careful with; 11 follow with practice which could be numerical indices practice for students with gaps. Use coaching worksheets and students in coaching pairs for depth and again to get students trained in a coaching atmosphere. Plenary: challenge task on PPT making up peer questions. WB9-11 rearranging algebra fractions to separate terms in index notation; with example where does not split up and examples with more than two steps or termsPlenary indices countdown with expressions. Identify any types of Qs students still struggling with.

BAT solve equations using the rules for indicesKeyQ: whats the inverse of a power;

A1014aWB 12-13

A1014bWB 14-18

Starter puzzle with differences of two squares. Then challenge question ‘what’s the difference’. Using indices with expressions to solve different types of quadratic equations. Use as AFL for quadratics. Answers to challenge at end of PPT. WB 12-13 solving equations by using inverse powers. Without calculators but encourage students to use calculators for checking solutions now and from now on. Followed by a set of practice questions including negative powers RAG. Followed by challenge questions – aimed at discovering and talking through misconceptions

WB14-15 rearranging expressions with powers to the same base numberWB 16-17 extend to solving equations by changing both sides to same base number and then equating the powers. Followed by practice set questionsWB 18 algebra question – rearranging to write in terms of given expressions

BAT simplify algebraic surds using (√ x )2=x ; √xy=√x √ y ; Key Q: what do we look for when simplifying √ xy ; when does the rule give us an integer value?

A1015aWB 19 -

Starter: odd one out with indices algebra RAGArea puzzles lead to students discovery of the rule √ab=√a ×√b . most should know this rule from gcse so check students understanding to judge pace to go through. Followed by exemplar answer showing how to use rule in reverse to simplify surds. Discuss how it works and methods for structuring answers. keyQ what is the least number of steps the examiner wants to see? Draw attention to√a×√a=a when it comes up. WB19 simplify surds using rule as set discussed. Followed by practice questions on PPT or worksheets

Connect 4 two player game on PPT – can give students copies of the grids. Practicing simplifying surds. Followed by more practice questions on PPT if needed.

With some examples of √ ab=√a

√b and multi-step problems the last of which leads

to add and simplify met in next lesson

BAT add and subtract with surds; Key Q: how can we use the rules to simplify calculations with surds

A1015bWB 20-21

Starter comparing surds RAG. Decide which is greater using √ab=√a ×√bWB adding surds by rewriting to same base root. Discuss why it works and extend as needed by modelling more examples. WB Now model example of perimeter around shape followed by few more examples on PPT Practice using set of questions on PPT or worksheets RAG.Notes – summary reminder of rules for surds met so far. Then challenge questions RAG. All three challenges are worth discussion and getting students to demonstrate answers to class. Answers to challenges are on the PPT.

BAT use surds in context of Pythagoras problems: links to finding distances met later in geometryKey Q: why is it useful to use exact answers in solving problems; how do we find the distance between two points;

A1016WB 22-25

Starter reminder of the definition of a surd as a+b√c then questions on simplifying surds RAGWB draw a line with length 5 – leads to using Pythagoras and exact surds to get an answer. Followed by example using Pythagoras with exact answer. Then extended to distance between two coordinate points. WB Pythagoras problems and practice first with finding missing lengths then finding areas of triangles. Check solutions with calculators but stress the value of exact answers.

BAT multiply with surds keyQ:

A1017WB 26-28

Starter: simplifying surds of the form √a ×√b particularly aimed at √a√a=a RAG. followed by challenge multiplication grids with multiplication and division using same rules. WB 16 full multiplication of surds (2−√3 ) ( 4+√3 ) using FOIL – relate to bracket expansion with quadratics. Simplifying the answer. Followed by practice questions on PPT RAG. Worth doing some more peer coaching using relevant worksheets – starter page on PPT. WB 27-28 problem solving questions at exam level. Use as AFL to check students misconceptions and if more practice needed before moving on to rationalising.

Add more problem solving questions for more able if more practice needed. Extension: The Tripods activity at back of PPT and on worksheet can be used for students to investigate and discover techniques with division by a surd as working backwards.

BAT use DOTs leading to rationalizing (√ x+√ y ) (√ x−√ y )=x− y ; Key Q: what operation rationalizes a surd; how can we use this to aid division by a surd;

A1018WB29

Starter simplifying surds of the form √a ×√b again RAGWB29a explore and discuss what multipliers will make a surd into an integer using starter questions as reference. Then WB 29b models multiplication by conjugate. Followed by set of practice questions on PPT RAG. Then summary notes for rationalising a surd – make sure students familiar with vocabulary.

BAT Use rules indices / Simplify and rationalise surds to solve problems or give proofs; Key Q: how do we do division with surds; what is a simplified surd;

A1019 WB 30-31

A1020

Starter rationalising surds using multipliers RAGWB30 introducing division by writing as a fraction with surd denominator. Then asking keyQ: how can we use a multiplier to change/manipulate this calculation? Followed by formal examples building up use of conjugate to more complex examples. WB31 medium difficulty example. Followed by lots of practice questions on PPT RAG. Worth doing some more peer coaching using relevant worksheets – starter page on PPT. Plenary: Could ask students to make up questions and try out on each other. See chili pepper challenge page on PPT At some point remind students that all the skills learned here are likely to be used in another context e.g. indices in solving equations; surds in finding distances… Extra plenary: Surds Bingo can be used as AFL to see which students need more support and if class ready to move on

BAT solve equations using the rules for indices and surds; Key Q: how do you deal with roots in equations, exact answers;

Will meet disguised quadratics later on after quadratics topic

A1021 WB 32-34

Starter rationalising surds RAGWB32 model a calculation leading to simplification, multiply by conjugate and cancelling. Writing the answer in the form a+b√c . could follow with students making up own questions. Aim is to build skills WB33-34 solving equations using skills built up with surds. Followed by challenge on PPT and worksheet. In the challenge each answer is used in the next question. Answers to challenge at back of PPT.

A101b QUADRATICS2.3 Work with quadratic functions and their graphs; use discriminant of a quadratic function, including the conditions for real and repeated roots; Completing the square; Solution of quadratic equations, including solving quadratic equations in a function of the unknown


BAT convert between completed square and normal formBAT rearrange and solve quadratics using completed square formKey Q: what connects completed square graphs with graphs of quadratics; how do we identify key points on quadratic graphs; what if the coefficient of x squared is greater than 1?

Probably 2 lessons

A1022a WB 1 to 2

A1022bWB3

Starter factorise and solve. Example to show that completed square rearranged is a normal quadratic. Show that completed square is transformation of x squared graph. Followed by practice rearranging into normal quadratic form. Could use opportunity here to sketch graphs using transformations or leave this until next part of lesson. WB1ab rearranging into C square form in steps with directions for notes. Then add sketch graphs. Annotate and discuss sketch graphs. For instance discuss the y intercept as well as minimum point. Note the distinction between even and odd coefficient of x – WB2 has odd coefficient leading to work with fractions. Followed by practice questions on PPT or worksheets. Encourage / expect sketch graphs and identifying the minimum point with each question.

WB2ab rearranging to completed square then solving using inverse operations in structured steps. Use sketch graphs to show roots and minimum point and discuss. Links to main method for solving equations, use extra annotation to show operations applied to both sides if appropriate. Discuss reasons for exact answers and calculated answers rounded to 2dp or 3sf. Followed by practice questions on PPT or worksheets. By now students should be able to produce sketch graphs if asked for them.

Starter factorise and solve RAG. WB3ab rearranging to completed square when coefficient x squared is not 1 – carefully model pulling out a numerical factor. Point out common errors such as forgetting to multiply back at the end and not giving the answer in the form required by a question. Then model solving from completed square form but note that the next lessons will show an alternative for tricky quadratics like these. Followed by practice questions on PPT or worksheets. Advice from examiners is to do lots of practice.

BAT manipulate quadratic expressions and solve using the quadratic formula; Key Q: what can we do to solve a quadratic that does not factorise or easily rearrange in completed square form

A1023WB4

Starter writing in completed square form RAG where green question is x2+bx+c discuss the solutions on the next page. Then model solving the equation = 0 to find roots. Leading to smaller version of quadratic formula. A challenge can be to derive the full quadratic formula but save time by briefly discussing and giving the formula. WB4ab using the formula to get the roots. Students often make mistakes with this in exams so structure solutions in clear steps and discuss common errors. Each example followed by a sketch graph – another chance to discuss exact answers v decimal rounded answers. Also some manipulation needed to put in surd form a+b√c . Do not leave surds in fraction form. Followed by practice questions on PPT or worksheets – include some examples which do not solve/have one solution and discuss when they come up.

BAT rearrange and solve disguised quadraticsKey Q: when is an quadratic not a quadratic; and when is it a quadratic? When can you factorise an equation using FOIL

Needs some dots practice - (x+ 1x )(x−1

x)

A1024WB5-7

Starter write in completed square and solve RAG. Solutions on PPTWB5 rearranging to a quadratic by multiplying throughWB6 rearranging by substitution of y squared with y WB7 rearranging by substitution of square root x squared with xNote that students could choose to actually use a substitution but I prefer keeping the equation as it is and using the structure of factorising into brackets. This allows direct route to solutions without having to substitute back.

BAT Sketch quadratic graphs showing intersections and max/min pointKey Q: what are the key features of quadratic graphs; how can we figure out the equation of a quad graph from its key points;

A1025

Starter identify the graphs – this can be done with geogebra instead of PPT questions. Given roots and / or min/max point, identify the eqn of the graph using factorising or completed square form. Several examples and activities on PPT modelling using roots and max/min point to determine equation of graph. This can be done with geogebra fairly easily and students can learn by exploring. Then can do graph sketching activities such as identifying / recreating graph pictures. If using Geogebra then can encourage students to make their own questions/ pictures for partners to recreate.Geogebra task – identify the quadratic from its graphGeogebra task – draw the picture of a set of graphs on same axes using Geogebra

USEFUL TO DO 101d INEQUALITIES HERE – PPT1026 PPT1027

BAT know how to use the discriminant to solve problems and understand properties of quadratics; Key Q: what is the discriminant and when is it useful? Investigation: activity sheet on investigating the discriminant is a useful way to introduce this as independent study.

NOTE – discriminant questions are often inequalities – good idea to do this topic after inequalities

BAT Sketch quadratic graphs showing intersections and max/min pointKey Q: what are the key features of quadratic graphs; how can we figure out the equation of a quad graph from its key points;NOTE – transformations of graphs met in 101ef but really worth practicing graph sketching until all students fluent with quadratic sketches at this point

A1028a investigate independently

A1028bWB 1-5

Set task to investigate discriminant of a selection of quadratics and find the rules connection to graphs of quadratics. Using a graph package such as Geogebra.

Starter Draw three graphs – one of each type of discriminantStudents use graph package to investigate in lesson or to illustrate findings from investigation. Next pages summarise the rules for the discriminant WB1 using the discriminant to decide the nature of the roots. Add more examples as needed. Can use geogebra to demonstrate solutions WB 2-4 discriminant algebra problems. Finding unknowns given conditions of roots and applying discriminant rules to build and solve equations/inequalities. These questions also bring together other features of quadratics and their graphs. Followed by further practice with worksheets or exam questions. Common errors to watch for are incorrect signs building equations and incorrect solutions to inequalities when students don’t use sketch graphs as supportWB 5 two sketch quadratic questions – checking that students have absorbed the relationships between forms of quadratic and graphs – ask students to do without graph package as mini test.

A101c EQUATIONS2.4 Solve simultaneous equations in two variables by elimination and by substitution, including one linear and one quadratic equation


BAT solving equations with ‘awkward numbers’ Key Q: how do the rules for solving equations work with fractions,

A1029 WB 1 -2

Starter examples of solving equations with integer coefficients RAG; students to discuss the ‘rules’ and decide what are common methods; including quadratics;

decimals, surd or trig coefficients … WB1 examples of non-integer coefficients, students should understand that the methods are the same however ‘more work’ may be necessary to keep solutions exact; discuss exact versus rounded answers – e.g. in 1b keeping the coefficients as surds leads to an integer solution – ‘what happens if we change the surds to decimals at the start?’. This is an opportunity to review learning on surds and introduce students to using exact trig values; students should find content challenging – if not ready for this content put off this lesson and use later/ have catch up work for students not ready WB2 revisits the quadratic equation and exact values; 2nd example with surd coefficients. Emphasise use of exact values; expect the unexpected from exam level questions … examiners are expected to make new problems in each exam

BAT know and use methods for solving simultaneous equations → by equating yKey Q: how do we solve equations with two or more unknowns;

A1030WB 3-5

Starter: solve simultaneous equations by elimination RAG The expectation is that students already know elimination method. If not they must learn this in independent studyWB3 solve quadratic by factorising. Quick review so that students WB4 four examples of solving simultaneously by equating y value. First with linear equations then with one linear and one quadraticWB 5 three further examples where the linear equation is rearranged before substitution. Followed by further practice on PPT or worksheetsMistakes are often due to signs errors or algebraic slips which result in incorrect coordinates. Students should be encouraged to check their working and final answers, and if the answer seems unlikely to go back and look for errors in their working. ACT geometric puzzle simeqns. Build solve problem from rectangle in circle. Worth using at an appropriate point for relating sim eqns to solving problems in context

BAT know and use methods for solving simultaneous equations → by substitution Key Q: how do we solve simultaneous equations when one or more equation is non-linear

A1031WB 6-7

Starter: build and solve algebra problems in words leading to algebraic formulation and solution by a substitution methodWB6 two examples of full on substitution method. Use colour to illustrate the substitution. WB7 two more examples, this time with equations with both x squared and y squared.

Followed by further practice on PPT or worksheets. Again encouraging students to check their answers. Some students could be encouraged to extend this to graphs which is met in 101e graphsNote the specification states: Students must be aware of the context and ensure that the solutions they give are appropriate to that context. Students should be aware that sim eqns can appear in most topics. Here we have the tools to solve them but later the context will matter as well.

Problem solving qs??? Or put with graphs Investigate when simultaneous equations cannot be solved or only give rise to one solution rather than two

BAT solve equations with indices; BAT solve simultaneous equations from index and graph problemsKey Q: how do we get simultaneous equations from index equations with two unknowns.

A1032 WB 8-9

WB8 Starter: solve index equations based on growth equations – without graphs (gcse level). Use as AFL review of indices. Extend with more equations with different base numbers if students need more practice WB9 build sim eqns from equating powers where base number is the same then solve simultaneouslyWB10 build sim eqns from equating powers where have to rearrange first to make base number the same, then solve simultaneously. Answers on next pages

GRAPHS and simultaneous equations follows in 101e

A101d INEQUALITIES 2.5 Solve linear and quadratic inequalities in a single variable and interpret such inequalities graphically, including

inequalities with brackets and fractionsExpress solutions through correct use of ‘and’ and ‘or’, or through set notation; Represent linear/quadratic inequalities graphically


BAT Solve quadratic and linear inequalities BAT solve inequalities problems in context Key Q: do we solve inequations the same way as equations; how can graphs help;

A1026a

A1026bWB1-2

True false set of questions for reviewing what students know. Extend the questions as go along.

Starter identify inequalities from number lines WB1 straightforward examples of solving inequations building up to brackets, x both sides and then negative coefficients of x. Try to demonstrate different methods using students solutions. WB2 examples of questions in context. Draw diagrams to help students see how

the equation is built. BAT Solve quadratic and linear inequalities BAT solve inequalities problems in context Key Q: how can sketch graphs help us solve inequalities;

Extension: opportunity here for graphical inequalities and simultaneous equations combined

A1027WB 3- 6

Starter solve basic inequalities with x squared RAG This gives a chance to show that there are two answers. Next pages show each of the solutions graphically - discuss – the solutions are the set of x values that work. WB3 two examples of quadratic inequalities. WB4 Extend with example of quadratic with negative coefficient x squared. Model two ways to deal with this – directly from graph or by multiplication through by -1 which changes the inequality sign. Emphasise using graphs to help get inequalities correct. Examiners comment on mistakes made with these so students need this hammered in to them. WB 5-6 solving linear and quadratic inequality and combining the solutions leading to similar question in context. WB7 extension question simultaneous equations with quadratics and graph sketching combined. Use geogebra to visualise the problem.

A101e GRAPHS 2.9 Understand the effect of simple transformations on the graph of y = f(x) including sketching associated

graphs: y = af(x), y = f(x) + a, y = f(x + a), y = f(ax)


BAT explore graph properties; Key Q: what can we find out about graph properties; what do we already know;

Starter: Students can be given sketches of curves or photographs of curved objects (e.g. roller coasters, bridges, etc.) and asked to suggest possible equations that could have been used to generate each sketch.ACT curve match. First pictures of curves from real life, discuss where we might need to use graph models in real life. Then task to match five equations and graphs. Hint for students = ‘try points’; ‘which graphs will be higher up for this value of x’; ‘which graphs are u or n shaped’. Discuss what is needed to

BAT explore transformations of graphs Key Q: what happens if we introduce a new constant to the equation of a graph, how will the graph and equation be affected;

A1033aWB 1-7 Starter: identifying quadratic graphs. Using roots, turning points, y-intercept.

Relate to shifts of y=x2 and describe these as transformations in x and y directions. Second page of quadratic graphs if needed for more consolidation. Could also extend this with peer questions or just put some more graphs up on Geogebra for students to identify

WB1-2 shifts demonstrated then example of quadratic followed by generalised graph shape. Explain the mark-scheme for this type of questions usually involves marks for each key point and a mark for correct shape. Most common error is translating the curve in the wrong direction for f(x + a) or f(x) + a. Encourage students to answer in bold colours so that their answer is clearly visible for later review. WB3-4 stretches demonstrated then two examples of generalised graphs. Most common error is apply the wrong scale factor when sketching f(ax). More examples can be used in which a graph is transformed by an unknown constant and students encouraged to think about the effects this will have. WB5 reflections demonstrated then example of reflection in y-axis. WB6 describe transformations in words – note that f(ax+b) not met yet. Check correct descriptions being used – full explanations rather than shorthand. Followed by: WB7 demonstrates using substitution to get the new equation algebraically after a given transformation. This is likely to lead to then finding key points on the new graph algebraically such as the intersections with axes. Workbook includes a set of notes on the six transformations for students to use as reference. Worthwhile to get them to colour the diagrams and check understanding by reading through at the end of the lesson.

BAT know and use the six types of transformations to graphsKey Q: can we get to some fluency with transformations of graphs;

A1033bA1033c

A1033d

1033b is a Peer questioning activity followed by Tasks investigating further for which there is a worksheet: ACT Geogebra challenge.1033c is a PPT with a set of questions and solutions covering most types of transformations of graphs Worksheets and further practice can include opportunities for students to make their own questions and explore links with graph properties and types of graphs with Geogebra – which are easier to explore now that students have the concept of transformations. PPT1033d basic review if needed for less able students

A101e GRAPHS 2.7 Understand and use graphs of functions; sketch curves defined by simple equations including polynomials, Interpret

algebraic solution of equations graphically;


2.8 use f(x) notation BAT expand brackets; divide algebraically by an exact factor using table method; explore and discover the factor theorem; Key Q: how can we find factors with non-linear equations

Formal method for algebraic division met later???

A1034WB 8-12

WB8 Starter: expanding one, two three brackets RAG. Can extend to asking what values give roots / zero. WB9-10 expand three brackets using a table followed by practice on PPT leading to WB10 two examples of factorising by reverse engineering using a table and more practice. ensure students are able to check their solutions by expanding. USE OF ICT: Use calculator table function to find integer roots of polynomials Followed by challenge activity consolidating factor theorem as method for establishing factors of polynomials. Could extend to quartic functions if appropriate as an extension. WB 11-12 factorising linked to the factor theorem which has been formally stated. Make sure emphasis is on clear working steps to show where the roots are – using calculator methods. Link this to graphs if appropriate or leave to next lesson.

BAT explore cubic, quartic and reciprocal graphs ; use f(x) notation; BAT explore graph properties – asymptotes and limits; Use factor theorem for finding roots and graph sketchingKey Q: what are the key features of other graphs; what are the main strategies for sketching graphs; what types of graph should we expect to meet in exams;

A1035a CUBICSWB 13-16

Starter identify graphs. Review of quadratic graphs. Discuss max/min; roots and y-intercept. Can use Geogebra for more/ alternative questions.Introduction to graph shapes – inform students that one mark given in exam Qs is often for correct shape of the graph. Then focus on cubic shape – discussion page on number of roots and task to sketch cubics with 1,2,3 roots – also discuss positive and negative cubic shapes. WB 13: sketch graph of cubic using table of values. Use task to check students substitution into function correctly. Page showing example of turning points – note differentiation not met in course at this point. WB14 transformations of cubics with key points. Note students should stick with single transformations at first. Can follow with more practice as needed.

When sketching cubic graphs, most students are able to gain marks by knowing the basic shape and sketching it passing through the origin. Recognising whether the cubic is positive or negative sometimes causes more difficulty. Students should have a ‘tick list’ of strategies for graph sketching;

WB15 Reminder of factor theorem and using calculator to find roots. Example of

identifying cubic graphs equation using the roots. WB 16 drawing cubic from the equation using the roots and positive or negative shape. Cubic and quartic equations given at this point should either already be factorised or be easily simplified (e.g. y = x3 + 4x2 + 3x) as students will not yet have encountered full on algebraic division or differentiation.

A1035b QUARTICSWB 17-19

Starter – identify how many roots different equations will have RAG. Discuss and AFL - check students understanding of quadratics and Cubic’s. WB 17 introduce Quartic’s with graph and then using calculator to find roots. This example f ( x )=(x−2)2(x+1)(x−1) has three roots. Discuss how many roots possible and different graph shapes. WB18 transformations of Quartic’s with given key points. Review and again stick to single transformations but can extend this to multiple transformations. WB19 sketch Quartic’s by factorising and then using roots first, then trying extra points to confirm shape. Note that these examples are generally with integer roots and straightforward factorisation. Example from specification is: Sketch the graph with equation y=x2 (2 x−1 )2

A1035c RECIPROCALWB20

Starter: find the y-intercept of each graph. Can extend to also finding x intercepts. WB 20 reciprocals: first introduction to shape of reciprocal graph then example of reciprocal function where asymptotes are x=0 and y = 2

WB21 transformations of y=1x . Can be extended to finding where graphs

cross the axes

WB22 Sketch graphs where two transformations of ¿1x . Specification states:

Students should be able to sketch curves like y= 2x−3

+2. Followed by asking

students to make up more examples and try out on each other

ACT identify bingo: matching equations to graphs. Shuffle the pages before

starting. There are four of each type: completed square quadratic; factorised cubic; factorised quartic; reciprocal with two transformations

BAT find intersections of graphs by solving equations; find new f(x) after a transformation algebraicallyKey Q: can we solve problems at exam level putting together all we know about graphs and solving equations;

Add sim eqns by graphs questions LOTS LOTS LOTS

Add exam q with f(x) +3 reciprocal and intersection axesOne for each type of graph?

A1036WB

Students should be encouraged to check any answers they have calculated against their sketches to check they make sense Links can be made with sketching specific curves. Students should be able to sketch curves like

y=(x−3)2+2 and y= 2x−3

+2

Students should be able to apply one of the six transformations to any of the

functions listed: quadratics, Cubic’s, Quartic’s, reciprocal, y=ax2 , sinx, cosx,

tanx, ex and ax growth functions. Specification does not say how much is expected with multiple transformationsStudents should be able to justify the number of solutions to simultaneous equations using the intersections of two curvesStudents sometimes fail to recognise the significance of a square factor in the factorised form of a polynomial.Extension: graphs with non-vertical asymptotes; explore graphs as x→ ± ∞

including y=ax+b

x2+cx+d

A102

A102a COORDINATE GEOMETRY - LINEAR2.7 Understand use proportional relationships and their graphs3.1 the equation of a straight line, parallel or perpendicular, Be able to use straight line models in a variety of contexts3.2 the coordinate geometry of the circle; properties: the angle in a semicircle is a right angle; the perpendicular from the centre to a chord bisects the chord; the radius of a circle at a given point on its circumference is perpendicular to the tangent to the circle at that point


BAT explore gradients of parallel and perpendicular line; rearrange and find equations of lines; apply conditions for parallel and perpendicular lines; Key Q: can we find gradient and y-intercept from equations of lines; what are the properties of parallel and perpendicular lines;

A1040WB1- 4

Starter is identifying equations of lines from their graphs using gradient and intercept. Opportunity for discussion and establishing what students know from GCSE. Identify which students need to go back to bridging topic for revision. To help students see how much information is given in the equation of a line, a good activity is to give an equation and ask students to find everything they know about that line, e.g. the intercepts, a point on the line, the gradient, a sketch, a parallel line, etc.Followed by notes which are in workbook on general equation and gradient and y-intercept. Then WB1 rearranging to give m and c . Then graph demonstration of parallel lines and two challenges rearranging to find lines with same gradient. Along with an example of graphical solution simultaneous equations – discuss /explain why rearranging equations is often used. Then step by step discovery of gradient property of perpendicular lines over several pages. WB2-4 with some practice of basic skills with gradient. Model solving basic problems – it’s useful to put the problems on Geogebra to demonstrate the solution.

BAT Derive and use y− y1=m(x−x1) from gradient between point and general point; find midpoints, distances and solve multi-step problems; solve linear geometry problems; Key Q: what quick method do we have for finding equations of lines;

(Two lessons) find distances and areas between points; Explore equations of lines, know the general equation of a line; find intersections;

A1041 WB5- 13

Starter finding gradient and intercept from equations. Can be extended if more practice needed at this point.Then the formula for distance between two points is derived building up from examples to algebraic form. Make sure students have ‘got it in words’ as memorising the formula is not necessary. Opportunity here for students to do this as a proof in steps. Some practice questions on PPTFollowing this there is derivation of the formula y− y1=m(x−x1) built from understanding of gradient as change in y over change in x. WB 6 finding line from two points using formula followed by practice questions on PPTWB7 finding lines activity – lines joining three points. This is an exploration into gradients and intercepts and useful; for assessment. Followed by challenge activity where students make their own question – students need guidance here so that lines are not too tricky or too easy but we want students to work with fraction gradients. WB8-13 first establish how to rearrange equation to given form then the questions go through finding lines; parallel and perpendicular lines; using the formulas. Can set these as a group of questions like an exercise so long as check that students have correctly completed all the problems

Alternative Formula y− y1

y2− y1=

x−x1

x2−x1 can be used. If used, care needs to be taken to ensure that the y-

values are substituted into the correct places and that negative signs are taken into account

BAT solve linear geometry problems;Key Q: what types of problems should we able to solve; can we learn by doing;

This is a set of extra problems for practice

A1042WB16-23

Starter finding midpoint of two pointsWB16 class discussion problem. Students will assume lines are at right angles – extend discussion by considering if lines are not perpendicular. WB17-23 further set of problems developing problem solving skills. Can set these as a group of questions like an exercise so long as check that students have correctly completed all the problemsWB24 problem to find shortest distance between two lines worth discussion in class. Then another good discussion activity is Venn diagram putting equations to meet given criteria. This can be extended by changing the criteria; getting students to set criteria; including the ‘none of these’ answer.

Students should be aware that DIAGRAMS are not to be relied upon and ‘spotting’ answers by looking at a diagram without providing evidence to support this will not gain full marks. However, candidates should be encouraged to use any diagrams provided to help them answer the question. Answers to length and distance questions are likely to be given in surd form, giving further practice in simplifying surds. Students should be encouraged to give answers in exact form unless specified otherwise. Students can be asked to find the area or perimeter of composite shapes.

BAT use linear graphs in modelling; Key Q: in what kinds of context will we meet straight lines; Real-life situations such as conversions can be modelled using straight-line graph and linked to ‘direct proportion’.

A1043WB25-26

Starter: finding equations of lines from two points. Involving fraction gradients and giving answer in form ax+by=c. Use as assessment to check student understanding. Discussion on what a mathematical model is are useful to prepare students for later work on mechanics and statistics. WB25 links direct proportion and linear geometry and also allows students to grasp how to ‘interpret’ the equation in a given context. WB26 is an example of real life context where the line does not go through (0, 0). Interpretation includes reliability of predictions from a scatter graph.

A102b COORDINATE GEOMETRY - CIRCLE 3.2 the coordinate geometry of the circle; properties: the angle in a semicircle is a right angle; the perpendicular from the centre to a chord bisects the chord; the radius of a circle at a given point on its circumference is perpendicular to the tangent to the circle at that point


BAT Find midpoints and distances; Solve problems with circle geometry using midpoints and distancesKey Q:

A1044 WB1-8

Starter finding midpoints leading to quick notes on finding centre of circle and radius using ‘what you already know’. WB1-4 basic problems on radius and points/centre on circle. WB5-8 extends to more complex problems. WB5 demonstrates example of a circle theorem -right angled triangle using diameter. WB6 perpendicular line to a diameter; another circle theorem: perpendicular to chord passes through a circle; WB8 using circle theorem from WB7 to find centre from two chords. Emphasise the problem-solving nature of questions. Using geogebra and drawing sketches as well as annotating given diagrams will help students to understand the question in many cases and so should be encouraged. Including find the coordinate of a point given the midpoint and one of the end points; find the perpendicular bisector.

BAT Find and use equations of circles; find the centre and radius of a circle from its equation

Key Q: what links the centre, radius and equation of a circle; how do we find where circle intersect lines;

BAT Find and use the general equations of circles; find intersections (circle and line); use properties of chords and tangents;

Two lessons

A1045WB9-11

A1046WB12-15

Circle theorems from GCSE (9-1) Mathematics can be used in questions so a quick recap could be useful and then they should be incorporated into questions. Examples of this include: finding the equation of the circumcircle of a triangle with given vertices; or finding the equation of a tangent using the perpendicular property of tangent and radius.

Starter centre, radius and equation from diameter AB Three pages notes on using equation of a circle. Basic run through of structure of equation of a circle. A good back up to this is the activity ‘Corners of a rectangle’. Add extra practice in finding centres and radius as appropriate; ensure students are drawing diagrams; The equation of the circle (x−a)2+( y−b)2=r2 can be derived from Pythagoras’ theorem, giving students the opportunity to look at proof.WB9-10 finding centre and radius from equation. Finding equation from centre and radius. Challenges on PPT and worksheets – finding equation of circles in pictures. Good Geogebra activity. WB11 problem show that radius is perpendicular to line (tangent) Followed by venn challenge matching equations circles to criteria.

Starter challenge activity changing one aspect (radius, centre) so that two circles touch Notes on general equation of circle. Go through carefully and relate to completed square.WB12 rearranging from general form to centre, radius form. WB13-15 intersections: circle and x axis; circle and line; circle and line with no solution; Students should be familiar with the equations x2+ y2+2 fx+2gy+c=0 and (x−a)2+( y−b)2=r2. ‘Complete the square’ method should be used to factorise the equation into the more useful form. Most

errors when completing the square to find the equation of a circle involve the constant term. Students may forget to subtract it or perhaps add it instead. Having found the equation, when giving the coordinates of the centre students must take care to get the signs the right way round as marks are easily lost by getting this wrong.

BAT solve geometry problems using your knowledge of circles

Key Q: can we solve problems at exam level; can we investigate further geometry problems and proof;

Needs 10 more problems addedNeeds 10 more problems addedNeeds 10 more problems added

A1047WB16-18

Starter – intersections: circle and axis; circle and line; circle and circleWB16-18 set of problems for students to work through with answers on PPT. useful for: consolidation and review; discussion where misconceptions; building up exam skills; should be followed by assessment task Can add challenge using investigation tasks and/or using more problem solving / STEP level geometry problems The conditions in which a circle and a line intersect can be investigated, with students justifying which will and will not intersect. Investigate finding the equation of a circle given 3 points on its circumference – will there always be a circle through a random three points? (can you give a counter-example?)

Simultaneous equations can be used to find the points of intersection between a circle and a straight line. Students can also be asked to show that a line and circle do not intersect, for which the discriminant can be used. Finding intersections with the axes should also be covered.When substituting into equations to find the intersections with axes, students sometimes substitute for the wrong variable, for example substituting y = 0 when trying to find the intersection with the y-axis. Another error is substituting the entire bracket (x – a) for 0 rather than just x.When finding the equation of a tangent to a point on the circle, typical errors are: finding the gradient of the radius; finding a line parallel to the radius; and finding a line through the centre of the circle.

A103 part 1 Algebra

A103ab FURTHER ALGEBRA1.1 Understand and use the structure of mathematical proof, proceeding from given assumptions through a series of logical steps to a conclusion; 2.6 Manipulate polynomials algebraically, including expanding brackets, collecting like terms and factorisation and simple algebraic division; use of the factor theorem


BAT manipulate algebra with some fluency; simplify algebra fractions using A1051 Starter expanding brackets up to three brackets. Extend activity with more

factorization; use algebraic division; Key Q: do we have a formal method for division with algebra; what is actually happening in the steps of algebraic division;

BAT use substitution to find values and transform functions;??? – belongs with transformations of graphs?

WB1- 5 practice and use expanding brackets treasure hunt activity if needed.Starts with examples of manipulating algebra 1) Factorise and simplify examples. This can be extended using bingo activity on PPT1050 2) breaking algebra fraction onto separate terms. This is an important skill for calculus problems later in the course so take time with discussion and extend with more examples if needed. Also discuss the case where algebra fractions do NOT break up –could ask students how they could prove this with a counter example

Then go on to division. Start with examples of long division numerically. Discuss until students can explain fully each step. Extend by asking students to set each other questions – check with calculators. Students should be aware that long division is not always the best or quickest method to use and sometimes results in some complicated algebra.WB1-5 step by step algebra division building up in difficulty. Only division by (ax + b) or (ax – b) will be required. Equations in which the coefficient of x or x2 is 0 for example x3+3 x2 – 4 or 2 x3+5 x−20 will need additional explanation and practice. No top heavy fractions yet but this could be met as an extension activity for more able .Followed by two sets of practice questions RAG on PPT. answers to these not yet on PPT

BAT use the factor theorem; factorise a cubic; Use factor theorem for finding roots and graph sketching – SEE A101ef

Key Q: can we use the factor theorem to factorise a polynomial and sketch its graph;

A1052WB6-11

The factor theorem can be introduced through investigation by substituting different values and checking against division to look for patterns.

Starter expanding brackets. Can use as AFL assessment. Identify if students have mastered expanding brackets.Then notes – introducing the factor theorem with examples. Discuss and make sure students relate factorising to finding roots. This has already been met with finding roots of graphs so students should know how to use table function on calculator to find roots but can remind them of this. WB6-7 When using the factor theorem, stress the importance of checking the value that is substituted; a common error is to use, for example, f(1) rather than f(–1). You should also emphasise the importance of fully factorising expressions, as a fairly significant number of students stop when they have reached one linear factor and a quadratic factor. WB7 is ‘Factor theorem can be used to find an unknown constant’. For example: Find a given that (x−2) is a factor of x3+ax2 – 4 x+6. Followed by step by step

algebra division by tabular method – students should understand that they must be able to do algebraic division by long division but tabular method is quick and leads to less mistakes with complex expressions. With two sets of practice questions on PPT answers to these not yet on PPTWB8-10 develops using factor theorem formally to sketch graphs. Using Geogebra to check solutions. Can be extended by asking students to make own question though students sometimes find making up this level of question difficult. ‘This is an excellent opportunity to review curve sketching by asking students to give a sketch following factorisation.’WB11 use as AFL / plenary exam question on factor theorem. Extend and check question by asking for a graph. ‘factor in all the solutions.’ Activity: a challenge activity on the PPT for more able students and/or extension.

BAT use the factor theorem; use the remainder theorem; Use factor theorem for finding roots and graph sketching – SEE A101ef

A1053WB12-17

Starter expanding brackets. Last chance to sort any misconceptions!Notes going through examples of remainder theorem. Substituting to find remainederWB12-13 finding remainders. Students can use calculators to check but should practice their arithmetic skills. WB14-16 is more of ‘Factor theorem can be used to find an unknown constant’. Including where two conditions can also be given in order to form simultaneous equations to solve. Make sure students do enough practice to be fluent in these types of problem as examiners have noted students are often making errors with signs; writing expressions where equations needed; substituting 1 instead of (-1) for example … followed by set of practice question on PPTWB17 use as AFL / plenary exam question

BAT construct proofs Key Q: can we be familiar with proofs involving odd and even numbers;

A1054 WB1-6

Starter is algebra fractions – simplify to show that LHS = RHSWB1-6 Proofs with odd and even. Model proof by exhaustion then examples of results (proof by deduction)with odd and even numbers using nth terms of even and odd numbers. WB 6 is proof by counter example.

BAT construct proofs Key Q: can we extend above to get fully structured proofs of given results, without missing steps;

A1055WB7-12

Starter is algebra identities – simplify/rearrange to show that LHS = RHSWB7-12 Proofs with numerical results. Including multiples and divisibility. Model proof by exhaustion then examples of results (proof by deduction)WB9 subtraction in algebra, using place value ideas

WB10 Euclids proof – needs lots of discussion for students to get it. Suggest students should go through it again out of lessons until they see the steps as this is a classic example of logical thinking in proof. WB11-12 counter examples and finding mistakes in steps of working. Students should understand an exam question can be structured this way.

BAT construct geometric proofs; other proofsKey Q: can we extend above to get fully structured geometric proofs of given results, without missing steps;

NOT MADE YET

A1056WBXX

XXXNOT DONE YET

A103 part 2 Binomial expansion

A103b BINOMIAL EXPANSION4.1 use binomial expansion for positive integer power n; link to binomial probabilities;


BAT write out in full a series expansion of a bracketKey Q: how does the formula for series expansion work; what are common mistakes and how can we avoid them;

A1057WB 1-4

Starter and notes is six page introduction using expansion of (3+x )n as model and increasing power successively while looking for patterns. This leads to pascals triangle and then to the nCr coefficients and formulaWB1-4 are full expansions using the expansion formula (a+b )n Using non-unitary coefficients of x from the first example. Followed by practice questions on PPT. Can extend with more examples and getting students to make (sensible) questions for each other. Could discuss ‘what if we want the terms up to x4 only?’ Students should initially be introduced to Pascal’s triangle, which can be used to expand simple brackets. Should BAT set out a problem in clear steps – using three colours until got the hang of the structure. Setting out work clearly and logically will be invaluable in helping students to achieve the final answer and also to spot mistakes if necessary. Don’t move on until all have grasped structure of series expansion

BAT find coefficients in a series expansion; understand and use the nCr formula; find the unknown coefficient of an expansion; use to make an estimate or find an exact numerical value;

A1058WB5- 9

Starter expansions of (a+bx )4 use as AFL to check students able to use formula competentlyFirst notes and examples demonstrating how the nCr formula calculates each

Key Q: can we use the expansion formula to find unknowns in algebra type problems;

coefficient. Using examples of full expansions WB5-7 finding coefficients using nCr formula and pascals triangles. WB8-9 exam questions with algebra problems inclusive.

Students will need to be familiar with factorials and the C rn❑ notation.

Marks are most commonly lost in exam questions because of errors in expanding terms. For example not including the coefficient when calculating, say, (ax)2; not simplifying terms fully; sign errors; and omitting brackets. Good notation will help to avoid many of these mistakes. When writing expansions which involve unknown constants, some students fail to also include the x’s in their expansion.When using their expansions to work out the value of a constant, a significant number of students do not understand that the coefficient does not include the x or x 2 part and so are often unable to form an equation in the unknown alone.The limitations of the binomial expansion should be discussed.Students should practice finding the coefficient of a single term, they should also be able to deal with setting up simple algebraic equations to find unknown constants.Use of the binomial expansion can be linked to basic probability and approximations

BAT find coefficients in a series expansion; understand and use the nCr formula; find the unknown coefficient of an expansion; use to make an estimate or find an exact numerical value;Key Q: what can we use the expansion formula for; use it to do powers of surds, to make estimates, to solve algebra problems …

Answers to do on PPTAdd 3more qs to assessment tasks

Needs 10 more problems addedNeeds 10 more problems addedNeeds 10 more problems added

A1059 WB10- xx

Starter is making estimates of calculations to demonstrate accuracy of using the binomial series compared to hand calculationsWB10 model using the expansion to do powers of surds;WB11-12 finding estimates from the expansion by using appropriate values of x; discuss the limitations of the examples. WB13-XX then solving ‘work back’ problems where (a+bx )n has unknowns from a,b, n and some terms coefficients in the expansion are known. Questions often go on to ask students to use their binomial expansion to evaluate a number raised to a power. For example, evaluating (1.025)8 by substituting x = 0.025 into an expansion for (1 + x)8. Students should be advised that simply using their calculator to evaluate (1.025)8 will gain no marks as it is not answering the question.

What follows: Negative and fractional coefficients in the series expansion; arithmetic series; geometric series; sigma notation; – year 13

A104

A104a TRIG RATIOS and GRAPHS5.1 Understand and use the definitions of sine, cosine and tangent for all arguments; the sine and cosine

rules; the area of a triangle in the form 12

absin C

5.3Understand and use the sine, cosine and tangent functions; their graphs, symmetries and periodicity


BAT review Sohcatoa; sine, cosine and area rules; understand the ambiguous case with the sin rule (with degrees as units )Radians are met in A206 Key Q: can we find missing angles and sides in triangles fluently; in all circumstances and in context;

1-2 lessons + lots independent study (gcse revision)

A1061WB1-9

SOHCAHTOA is not covered here – only the advanced trig rules. Worthwhile for students to have tasks set prior to lessons to review GCSE content. Many will be comfortable with all the GCSE content and can be directed straight to challenge activities → questions should now be focused more on multi-step problems and questions set in context. Links to proof can be made, for example proving the area of a triangle; prove the sin or cos rules. Starter Sohcahtoa questions. AFL - Identify if students need support.WB1 -5 Notes on sin rule and cos rule including a proof of the cos rule that can be missed out / set as independent study task. Basic questions leading up to multi step questions. Plenary question – ‘The x and y coordinates of points on the unit circle can be used to give cosine and sine respectively’.WB 5 is 2017 GCSE proof questionWB6 -9 reviews area rule and builds up to more multi step questions

Students should be encouraged to write down any formulae they will be using before substituting in the numbers. Students should be able to solve questions in various contexts; these could include coordinate geometry or real-life situations. Questions may involve bearings, which may not be well remembered from GCSE so should be reviewed. Students should be encouraged to check that their answers are realistic as this check can show up errors. When completing multi-step questions emphasise to students that they should show all working out and use the answer function on their calculators to avoid rounding errors. It can be a useful teaching point to divide the class asking one side to round all answers and the other to keep values stored in their calculator to show how this affects the final answer.

A frequently seen error in these questions is students using the cosine rule to calculate an incorrect angle, sometimes despite having drawn a correctly labelled diagram. This indicates a lack of understanding of how the labelling of edges and angles on a diagram relates to the application of the cosine rule formula.

BAT know the key features of trig graphs using both degrees and radians; BAT use exact values of trig functions to solve problems; use the sine, cosine and tangent functions; their graphs, symmetries and periodicity

Key Q: can we solve problems with exact trig values; can we can we solve basic trig equations fluently using graphs to ensure the correct number of solutions;

A1062WB 10-14

StarterWB10 introduces the trig graphs – ask students to describe the features of the graphs and go through as class. Discuss symmetries; key points; shape; period of each graph; …Use of the graphs can be linked to modelling situations such as yearly temperatures, wave lengths and tidal patterns. The unit circle can again be used to show how the trigonometric graphs are formed. Characteristics such as the period and amplitude should be discussed. https://www.geogebra.org/m/S2gMrkbD

WB11 continues with symmetries of the graphs. Add more examples until students have got the different types. Followed by formal identities from the symmetries such as sin (−θ )=−sin (θ ). Opportunity to discuss odd and even graphs here. WB12-14 derive the exact values using SohCahToa and fill in results table. Students should understand that 1) they can get the exact results on calculator 2) they need to be able to work with the exact valesus. So WB13-14 give examples of this using symmetries and manipulating surds for proofs. Then typical exact value question met in mechanics problems.

ACTIVITY Exact values Bingo useful for reinforcing knowledge

BAT recognise and sketch transformations to trig graphsKey Q: can we apply what we know about transformations to trig graphs;

A1063 WB15-16

Starter – questions to review key features of sin cos tan graphsTransformations 1-6 pages of example on PPT modelling the different types of transformationsACTIVITY – give students sin, cos tan graphs on paper or whiteboards. Have a set of cards with single transformation and ask students to apply to one of the graphs. Checking answers on geogebra. Extend by asking students to do multiple transformations – will need to explain the ‘order of operations’ some of these such as sin(4 x+60)=sin ¿ Knowledge of graphs of curves with equations such as y = sin x, y = cos (x + 30), y = tan 2x is expected’WB 15-16 identifying transformations ACTIVITY – give students sets of graphs and ask them to identify the transformations –

https://www.geogebra.org/m/S2gMrkbD

again checking using geogebraCould ask students to make their own notes / select some questions and add them to their notes from the activities

A104a TRIG Identities and EQUATIONS

5.5 Understand and use tanθ= sin θcosθ ; Understand and use s∈¿2 θ+cos2θ=1¿

5.7 Solve simple trigonometric eqns, given interval, including quadratic eqns in sin, cos and tan and equations involving multiples of the unknown angle


BAT solve trig equations within a given intervalBAT derive the trig identities; use them to rearrange and solve equations; Key Q: can we solve harder trig equations fluently using graphs to ensure the correct number of solutions; what are the trig identities and how are they useful;

2 lessons

Solutions to practice Qs to do

A1064 WB17-20

Starter simple calculations with exact trig values RAGTrig identity pages 1-3 derives the two identities in steps with diagrams. Emphasise that students must know these and may have to use them in simple proofs.WB17-18 simple rearranging to give results using the identities. Again relate these to proof.WB19-20 revisits exact values using the identities

Solutions to practice Qs to do

A1065 WB21-28

Starter sketch graphs trig transformations RAG discuss and identify misconceptionsWB21-24 model basic trig equations with no transformations, finding the solution set and discuss the infinite set of solutions for θ ϵ R . Followed by practice questions RAG. Make sure students using diagrams with their solutions and using clear steps to consider the number of solutions in the given domain. Keep looking at the infinite set of solutions to remind students they need to get all possible solutions. Also - check solutions with calculators.“When solving trigonometric equations, finding multiple values within a range can initially be illustrated using the graphs of the functions. The decision can then be made whether to move on to using CAST diagrams or continue using graphs. Whichever method is used students will need plenty of practice in identifying all values within the limits

correctly. Intervals with negative solutions as well as positive solutions should be used. Students should be able to solve equations such as sin (x + 70°) = 0.5 for 0 < x < 360°; 3 + 5 cos 2x = 1 for –180° < x < 180°; and 6 cos 2x + sin x − 5 = 0 for 0 < x < 360°,

WB25-28 model harder problems with sin (ax+b ) ,cos (ax+b) , tan(ax+b) using same techniques but more careful final steps to find all solutions – CAST diagrams may be useful for these but be careful to avoid confusion. Sticking to sin, cos, tan graphs alone – make sure students write solutions in sets/as a list, before using inverse operations with (ax+b) for final solutions. Could ask students to research CAST diagrams as extension task. Followed by more practice questions RAG. Make sure students using diagrams with their solutions and using clear steps to consider the number of solutions in the given domain. Keep looking at the infinite set of solutions to remind students they need to get all possible solutions. Also - check solutions with calculators. Common errors include: not finding values in the given range; finding extra, incorrect, solutions; not going on to find the values of x and instead leaving the values for, say 2x or x + 30; algebraic slips when rearranging the equation; and not giving answers to the correct degree of accuracy. The loss of accuracy in the final answers to trigonometric equations is common and often results in the unnecessary loss of marks. Sketches of the trigonometric functions are often helpful to check all solutions have been found.

BAT derive the trig identities; use them to rearrange and solve equations; BAT solve disguised quadratic equations with trig functionsKey Q:

Solutions to practice Qs to doAdd more questions – variety / different types if possible

A1066WB29 –34

Starter: sketch more graphs trig transformations RAG discuss and identify misconceptionsWB29-34 model solving equations using identities to rearrange to tan or to quadratics. Using graphs to ensure correct number of solutions in the domain given. Can use Geogebra for checking using ‘putting one graph equal to another’ method i.e. graph both LHS and RHS and find x value of intersections. This will then develop ideas around transformations and properties of graphs.Students should be comfortable factorising quadratic trigonometric equations and finding all possible solutions. It should be noted that in some cases only one of the factorisations will give solutions but in most case there will be two sets of solutions. Situations where one answer is equal to zero can cause some confusion with students then not looking for further solutions. This sort of example should be covered in class. For example, the

equation, sin θ (3 sin θ + 1) = 0 will often be simplified to just 3 sin θ + 1 = 0, resulting in the loss of solutions to the original equation

What follows: small approximations; radians; arcs and sectors; other trig functions; trig identities and proof; double angle formulae; problems in context (mechanics) – in year 13

A105

A105a VECTORS 10.1 use vectors in two dimensions10.2 calculate the magnitude and direction; vector addition and multiplication by a scalar; geometrical interpretation


BAT use vectors in two dimensions; Key Q:

PPT WB

Students need to be familiar with column vectors and with the use of i and j vectors in two dimensions.Students should be able to find a unit vector in the direction of a, and be familiar with the notation |a|.The triangle and parallelogram laws of addition should be known and students should be able to use them. Students should understand that vectors are commutative.Students sometimes make mistakes when manipulating vectors in i and j form and should be encouraged to use column vectors when possible.

BAT calculate the magnitude and direction of a vector and convert between component form and magnitude/direction form; Key Q:

PPT WB

Where answers are given in surds they should be simplified if possible.Students should understand and be able to use the conditions for parallel vectors.Use the classroom floor as a 2-dimensional grid to help students visualise vectors. Use the position of students in the room to illustrate concepts.Consider vectors in the real world, e.g. ask students to think of everyday phenomena that

have a magnitude and direction e.g. forces, velocities, displacements.BAT add vectors diagrammatically and perform the algebraic operations of vector addition and multiplication by scalars, and understand their geometrical interpretations.

Key Q:

PPT WB

When performing operations on vectors this should also be understood geometrically, diagrams will be helpful here. Students should be able to use given diagrams but also draw their own in order to assist with questions.

BAT Key Q:BAT Key Q:

A105b VECTORS 10.3 Add vectors diagrammatically and perform algebraic operations vector addition and multiplication by

scalars, and understand their geometrical interpretations10.4 use position vectors; calculate distances between points represented by vectors10.5 solve problems in pure maths and in context


BAT understand and be able to use position vectors; BAT calculate the distance between two points represented by position vectors;

Key Q:

PPT WB Students should know and be able to use O⃗B−O⃗A= A⃗B=b−a

Students should be able to calculate the distance between two points (x1, y1) and (x2, y2) using the formula d2=( x1−x2 )2+( y1− y2)2

BAT use vectors to solve problems in pure mathematics and in context, (including forces;

Key Q:

PPT WB

Use the ratio theorem to find the position vector of a point C dividing AB in a given ratio.Use familiar shapes to illustrate the difference between 2 vectors and vector addition, e.g. parallelogram, rectangle.When solving problems using vectors only pure contexts are covered. Examiners comment that students understand the simple basics of vectors but are unable to deal with the complexity of ratios. Students should be given plenty of practice in identifying points that divide line segments in a particular ratio both externally and

internally

What follows: 3D vectors in year 13

A106

A106a DIFFERENTIATION7.1 derivative of f(x) as gradient of the tangent to the graph of y = f(x); the gradient of tangent as a limit;

interpret as a rate of change; gradient graph; second derivatives Differentiation from first principles for small positive powers of x7.2 Differentiate xn, for rational values of n, and related constant multiples, sums and differences

6+6 hours guided learning time

BAT explore differentiation and the gradient function of curves; differentiate from first principles; Key Q: what connects the gradient of a graph and the equation of a graph; is there a gradient function;

A1080ACT1 worksheet

Starter – sums with fractions. Useful for later algebra manipulationACT1 gradient function groupwork.Gradient function investigation. Introduce using notes to build idea of gradient of chord between two points approximating gradient of a tangent. Students in groups find gradients of points on y=x2 and y=x3 extend to other graphs if time (unlikely). Summarise and discuss results leading to differentiation rule.

BAT explore differentiation and the gradient function of curves; differentiate from first principles; Key Q: can we use a proof to get the gradient of a function – proof

from first principles; can we be fluent in using ddx

(a xn ) for single

terms;

A1081WB1-4

Starter – sums with expressions. Useful for later algebra manipulationNotes pages building up to first principles using graph explanation WB1-2 first principles step by step worked examples. Discuss carefully idea of limits and change in y values over change in x values.

WB3 finding the rule for ddx

(a xn ) from examples

WB4 set of one term examples. Go through and add more examples until students clear on the rule.ACTIVITY Bingo differentiation single terms Students should know how to differentiate from first principles. Students should be able

to use, for n =2 and n = 3, the gradient expression limh→ 0 ( ( x+h )n−xn

h ). The alternative

notations h → 0 rather than δx → 0 are acceptable.

BAT differentiate polynomials and find the gradient of curves Key Q: can we manipulate algebra to get a functions in polynomial

form so it can be differentiated; can we be fluent in using ddx

(a xn ) for

any terms;

A1082WB5-9

Starter WB5-6 model using the rule with single terms then multiple terms. WB7-9 focus on fraction and negative powers. Then expanding brackets to get separate terms. Then separating algebra fractions into separate terms – worth discussing the case

xx+1 which does not separate and cannot be differentiated (yet). Also worth

emphasising clear labelling of work, especially RHS of steps. Students often confuse their

own work mixing up between f(x) and dydx

steps when they are not labelled clearly.

Followed by Practice questions on PPT. These extend to substituting values into the gradient function. find derivative of polynomials – several terms, fraction and negative indices, algebraic fractions; Mistakes are easily made with negative and/or fractional indices so there should be plenty of practice with this. Students will need to be confident in algebraic manipulation of functions to ensure that they are in a suitable format for differentiation.

Common errors are with differentiating fractional terms such as 8x

;

BAT differentiate polynomials and find the gradient of curves Key Q:can we apply the gradient function to solve problems such as finding all points where the gradient is two on a graph;

A1083WB10-15 Starter finding the values of y and

dydx

at given values of x.

WB10 differentiation polynomials with negative and fraction powers as review previous learning. Following previous lesson there are ore practice differentiation questions if needed at start of PPT. Wb11-14 solving gradient equals value to find points on a graph leading to solving gradient function equals value to find two or three points on graph by solving quadratic or cubic. WB15 leading towards stationary points with an example of finding where gradient is zero.

BAT determine the coordinates and nature of stationary points A1084 Starter differentiation medium grade exam question. Rearrange to get powers,

BAT find the gradient of a function at a point and understand increasing and decreasing parts of functions; Sketch the gradient function for a given curve + identify where graph is increasing/ decreasingKey Q:

needs more Qsneeds more Qsneeds more Qsneed Q on sketching gradient functions of curves

WB16-19 differentiate and solve gradient equals zeroTwo pages notes demonstrating changing gradient on a curve. Also use geogebra stationary points for demonstration and discussion leading to definition and types of stationary points. WB16 model using stationary points to sketch graphs of functions . Determining turning points using rough gradient tables. Link back to sketching graphs using factor theorem. Followed by set of basic practice questions,WB18-19 more problems approaching exam question standard, finding stationary points by looking at gradient around points where gradient equals zero. Followed by practice questions. Practice 2 needs improvedUnderstand graphical explanations and apply understanding to solving problems – using sketches; Students should be able to identify maximum and minimum points as points where the gradient is zero.

BAT Find the second derivative of functions and use it to determine the nature of stationary points; Key Q: what types of stationary point are there and how do we find them; how do we determine their nature;

needs more Qsneeds more Qsneeds more Qs

A1085WB20-22

Starter draw three graphs and look for connections (function, derivative and 2nd derivative). Available on Geogebra. Leads to discussion on gradient of the gradient which can then be shown for a local minimum. Establishing the results for 2nd derivative. WB20-21 finding stationary point and determining its nature using the results. Followed by practice questions. Practice 1 = finding the 2nd derivative. Practice 2 = finding nature of turning points. WB22 exam level question. For AFL to check where class is at. Solve gradient functions f’(x) = 0. Use gradient tables to determine nature of points; use 2nd derivative to find nature of points. When finding a stationary point, some students use inequalities as their condition rather than equating their derivative to zero. Another error is to differentiate twice and solve f (x) = 0. ′′

BAT find equations of tangents and normals to curves; max and min; Key Q:what are tangent and normal lines; what links them and how do we find their equations; can we solve geometry problems involving tangents and normal;

A1086WB23-31

Starter finding equations of lines and perpendicular linesWB23-25 model finding equation of tangent line and normal line using point and gradient. Use Geogebra to demonstrate – can use the geogebra file to move the point and see the tangent and normal move accordingly. WB 25 is a second example of same with harder function. WB26-29 set of problems on tangents and normal. Reviewing geometry and differentiation using problems.

WB30-31 exam standard questions. Use as AFL to check progressThis reviews and extends the earlier work on coordinate geometry. Maxima, minima and stationary points can be used in curve sketching. Problems may be set in the context of a practical problem. This could bring in area, volume, trigomometry from GCSE. When working out the equations of tangents and normal, some students mix the gradients and equations up and end up substituting in the wrong place.

A106b DIFFERENTIATION7.3 gradients, tangents and normals, maxima and minima and stationary points; Identify where functions are increasing or decreasing


BAT understand gradient of tangent as a limit and its interpretation as rate of change; apply differentiation in context of mathematical modelling Key Q: what is the nature of the gradient of a function at different x values;

A1087WB1-10

Starter review gradient, gradient at a point, tangent equation RAGNotes on the difference between increasing and decreasing functions thenWB1-2 examining the gradient graph for cubic functions. Discuss the key features and careful explanation of links between and differences. Such as the stationary points of f(x) are where the gradient is zero. On the gradient graph the max/min points are where the gradient changes WB3-8 model finding values of the domain where f(x) is increasing or decreasing between turning points and asymptotes. (example of inflection point is in the assessment tasks). Using gradient tables and manipulating the f(x) and f’(x) algebraically. WB9-10 are modelling questions in context of velocity and acceleration. Finding values on graphs of f(x) and f’(x) and explaining the context and assumptions in the model. Understand gradient as rate of change, e.g. speed on a s-t graph; acceleration;

BAT apply differentiation to solve optimization problemsBAT manipulate algebra to form equations that can be differentiatedKey Q:can we find the max or min solution to a problem in context; what algebra manipulation can we do to take equations in two variables and substitute/rearrange to equations in one variable;

A1088 WB11-15

ACTIVITY – max box if time or as self study task. Starter algebra manipulation RAG. Simplify expressions similar to those found in optimisation problemsWB11-15 model optimisation problems building up in difficulty to exam level questions. Make sure students understand the two part nature of questions:

At least two lessons part 1 usually involves algebra manipulation substituting on e equation to another to get equation in one variable. Step 2 being to find the nature and position of turning points. Followed by practice questions on PPT or any set of exam questions. Structure working in clear steps; BAT explain why 2nd derivative gives max/min referring to graphs of f(x), f’(x) and f’’(x); use sketches of graphs in solutions to problems; Fluency in algebra manipulation for show that optimisation problems When applying differentiation in context, students should be ensure they give full answers and not just a partial solution. For example if asked to find the volume of a box they must not stop after finding the side length.

What follows: trig functions; exponential and logarithm functions differentiation in year 13. Chain rule; product rule; quotient rule; parametric eqns; implicit differentiation; growth functions

A107

A107ab INTEGRATION8.1 Know and use the Fundamental Theorem of Calculus (integration as reverse of differentiation) 8.2 Integrate xn (excluding n = −1), and related sums, differences and constant multiples8.3 Evaluate definite integrals; use a definite integral to find the area under a curve


BAT Integrate polynomial functions using the reverse process to differentiation Key Q: if we know the gradient function can we find the original function; can we match tangent eqn, normal eqn and gradient

A1120WB1

ACTIVITY https://undergroundmathematics.org/calculus-of-powers/tangent-or-normal/pairs-problem WB1 goes through this activity. Can use the PPT or website. Big review of tangent

https://undergroundmathematics.org/calculus-of-powers/tangent-or-normal/pairs-problem

https://undergroundmathematics.org/calculus-of-powers/tangent-or-normal/pairs-problem

function together; and normal: Matching normal and tangent together by analysis of gradients; then finding intersection point by simultaneous equations; then matching to given gradient functions by finding gradients at each point.Then the important bit: explore if we can decide what the original function was for each matching. Letting students explore the reverse of differentiation and find the rule independently. With a range of functions. Can extend this just by giving more gradient functions and asking what the ‘y function’ would be.

Integration can be introduced as the reverse process of differentiation. Similarly to differentiation, students should be confident with algebraic manipulation. Students sometimes have difficulty when integrating expressions involving negative indices.

BAT Integrate polynomial functions using the reverse process to differentiation Key Q: what rule can we use for integration of polynomials; what is the constant of integration and why does it exist;

Complete set of practice questionsComplete set of practice questionsComplete set of practice questions

A1121WB2-5

Starter differentiate two functions RAG showing that the gradient function is the same with any constant value term at the end. Make sure students have spotted this. Then find the original function from gradient functions (with same functions in a different order) leading to discussion about the constant of integrationWB2-3 models how a family of curves will have the same gradient function. Can be extended with other examples on Geogebra. Then introduce the correct notation for integration using the three functions from the starter

WB4-5 models formulaic approach to integration using rule y=an

xn+1+C. Building up

to several terms, terms with fraction and negative powers and algebra fractions. Lots of practice needed. Some practice questions on PPT. Students need to know that for indefinite integrals a constant of integration is required. Forgetting to add + c when working out indefinite integrals is a very common mistake and loses a mark.

BAT Integrate functions in context; find the value of + C Key Q: can we find the specific function through a point by integration; can we solve problems at exam level using calculus and geometry;

A1122 WB7-13

starterWB7-9 Discuss the family of curves produce by integration. Then model finding the value of C using a point. Emphasise the structure of the solution and labelling each step with y or dy/dx … WB10 model example of finding function from 2nd derivative. By successive integration and finding both constants of integration using given values. Could extend this with more examples.

WB11-13 model exam level questions. Rearranging equations, integrating and finding value of C. Extended by geometry problems like finding tangent and normal equations. Including integrating the second derivative and geometry problems; Given f (x) and a ′point on the curve, students should be able to find an equation of the curve in the form y = f(x).

BAT evaluate definite integralsKey Q: can we get a numerical value as the answer to an integration;

A1123 WB1-5

Starter integration of polynomials RAG. review and check able to udse the rule for integration. Add more questions if neededWB1-5 introduce definite integrals as a formula and model finding values step by step. Discuss what happens with the constant of integration. Develop with more problems as class together then use practice questions to consolidate. Three sets of questions RAG on PPT. Should be able to work with exact values – fractions and surds. It is important that students show their working out clearly as mistakes are easily made when putting values into a calculator. Students should also be encouraged to check their answers. Calculators that perform numerical integration can be used as a check, but a full method will be needed. Arithmetic slips are also a common cause of lost marks, often when negative numbers are substituted and subtracted after integration.

BAT estimate the area under a curve using the trapezium rule Key Q: can we estimate the area under a curve

OPTIONAL this topic can be left until A2. BUT it makes a good introduction to areas under curves – it is not assessed in A1 If using it – you can do this topic before, after or in between other integration topics

A1128WB1-5

Starter 1) finding areas of trapeziums. 2) Alternatively use a GCSE exam question on areas under curves or 3) https://undergroundmathematics.org/introducing-calculus/approximating-areas approximating areasThen notes on deriving the trapezium rule. Students do not need the proof but it helps them to use the rule efficiently if they can. W1-5 set of questions with varying number of strips and varying y equations. All questions in this topic are essentially exam level questions. ACTIVITY https://undergroundmathematics.org/introducing-calculus/underneath-the-arches explores over and underestimates

BAT use integration to find the area between a curve and the x-axis; evaluate areas under the x-axis; use integration to find the area between a line and curve or between two curves; Key Q:can we understand integration as the area under a curve

A1124WB6-12

Starter finding definite integrals RAG. Review and check able to use the rule for integration. Add more questions if neededWB6 introduce the idea of area under a straight line graph y = mx. Show that the definite integral from 0 to b has the same area then extend to the area between a and b and show this is the area up to b subtract the area up to a. Then discuss the area as strips of

https://undergroundmathematics.org/introducing-calculus/underneath-the-arches

https://undergroundmathematics.org/introducing-calculus/underneath-the-arches

https://undergroundmathematics.org/introducing-calculus/approximating-areas

width dx summed to make the area and relate to ∫a

b

f (x )dx Set

WB7-9 develop idea of area under curves with more examples. WB10-12 develop problems with areas under curves. First show that these are negative then model examples where the areas need to be calculated separately Work out problems as area puzzles with clear diagrams. Students will be expected to understand the implication of a negative answer from indefinite integration.Links can be made with curve sketching in questions where students need to find the points of intersection with the x-axis for a curve in order to find the limits of integration

BAT use integration to find the area between a curve and the x-axis; evaluate areas under the x-axis; use integration to find the area between a line and curve or between two curves; Key Q: can we solve area problems using integration; including the area between a curve and a line

A1125WB13-14

Starter solving simultaneous equations with curve and line or circle and line. Ask students what areas we could find in each diagram and howNotes describe how to combine integrals into one but students can solve problems in a variety of ways. They should be aware of quick ways of finding areas – triangles and trapeziums. WB13-14 model finding areas by combining the equations. Then short set of practice questions. There is a geogebra file with each question including the three practice qs.

BAT use integration to find the area between two curves; Key Q: can we find areas between two curves

A1126WB15-

Starter WB15 model finding areas by combining the equations. Then short set of practice questions. There is a geogebra file with each question including the three practice qsACTIVITY meaningful areas https://undergroundmathematics.org/calculus-of-powers/meaningful-areas finding areas between curves probably suitable for upper ability students only. Shows how some area can be found by symmetry or other geometrical technique alongside integration methods.

Starter ACTIVITY https://undergroundmathematics.org/introducing-calculus/speed-vs-velocity thinking about links between v-t and s-t graphs

BAT use integration to find the area between a curve and the x-axis; evaluate areas under the x-axis; use integration to find the area between a line and curve or between two curves; Key Q: can we apply integration to solve problems in modelling real life situations; can we apply integration to v-t and a-t graphs to find distances and change in speed;

PPTWB

https://undergroundmathematics.org/introducing-calculus/speed-vs-velocity

https://undergroundmathematics.org/introducing-calculus/speed-vs-velocity

https://undergroundmathematics.org/calculus-of-powers/meaningful-areas

Modelling could be left to A2 but there is a lot of integration in A2 so better if met here to start withWhat follows: Integration by substitution; integration by parts; differential equations – separation of variables; parametric equations; trapezium rule; exponential and logarithm functions;

A108

A108a LOGARITHMS and EXPONENTIALS6.1 Use the function ax and its graph, where a is positive; Know and use the function ex and its graph6.2 the gradient of ekx is equal to k ekx and hence understand why the exponential model is suitable in many applications6.3 l oga x as the inverse of ax, where a is positive and x≥ 0; the function ln x and its graph; use ln x as the inverse function of ex

6.4 Understand and use the laws of logarithms6.5 Solve equations of the form ax=b6.6 use log graphs to estimate parameters in relationships of the form y=a xn and y=k bx, given data for x and y6.7 use exp growth and decay, use in modelling (radioactive decay, drug concentration decay, continuous compound interest); consider limitations of models


BAT recognize key features of exponential graphs and logarithmic graphs; explore ex and its inverse ln x graphically; Write numbers and expressions as logarithms; use logarithmic graphs to estimate parameters in relationships of the form y=a xn and y=k bx, given data for x and y; Use exponential growth and decay in modelling, giving consideration to limitations and refinements of exponential models.Key Q: what is a growth function; what is a logarithm; how do we use logarithm functions with calculations;

A1100WB1-4

Starter introduction ACTIVITY. Discuss how growth function used as richter scale for earthquakes. Then investigate growth function y=2x finding values in real life context. Let students use phones for researchWB1-2 graphs of growth and decay. Can be explored further using geogebra. WB3-4 graph of logarithm functions. Discuss logarithm as inverse of corresponding growth function. Can be extended with transformations of graphs of y=ax and y=log x . Followed by WB4 solving simple equations – ask students to try values they think work and check with calculator. Looking for links and beginning understanding calculations before next lesson ACTIVITY – investigation https:// undergroundmathematics.org/exp-and-log/see-

https://undergroundmathematics.org/exp-and-log/see-the-power/problem


the-power/problem into y=10x– exploring the inverse i.e. the logarithm by pattern spotting

When sketching the graph of axstudents should understand the difference in shape between a < 1 and a > 1. Explain to students that ex is a special case of ax. Graphs of the function ex should include those in the form y=eax+b+c. Students should realise that when the rate of change is proportional to the y-value, an exponential model should be used.Errors seen in exam questions where students have to sketch exponential curves include: stopping the curve at x = 0; getting the wrong y-intercept; and believing the curve levels off to y = 1 for x < 0.

BAT apply the laws of logarithms and indices with numbers and expressions; Key Q: what are the laws of logarithms that correspond to the laws of indices;

A1101 WB5-10

Starter graph sketchingNotes on defining logarithm functions as log an=x means that n=ax then WB5-7 numerical examples. Building an understanding of the relationship between growth function and logarithm as inverses. Lots of questions can extend with work in pairs challenge – make a question for your neighbour.WB8-9 reversing the link to go from index numerical identity to logarithm and checking and practice ACTIVITY matching base numbers to make logarithms correct. WB10 ACTIVITY investigating addition of logarithms leading towards the addition law. choose integers 𝑥 and 𝑦 so that log6 x+log6 y=1 gives that xy=6 some students may be able to deduce the addition law from this.

A1102WB11-16 Starter finding x in log3 x=2; log3 27=x; log3

181

=x

Notes giving the laws of logarithms then quick proof of the addition law. Can be extended by asking students to research or create proofs of the other laws. Proof ACTIVITY on https://undergroundmathematics.org/exp-and-log/proving-laws-of-logs WB11-12 applying the laws to simplify calculations. Lots of questions and some more practice. Move on when students ready – can extend with work in pairs challenge. WB13-14 takes this on to simplifying algebraic terms then WB 15-16 are exam level questions.

https://undergroundmathematics.org/exp-and-log/proving-laws-of-logs


When using laws of logs to answer proof or ‘show that’ questions, students must show all the steps clearly and not have jumps in their working out.

BAT solve equations of the form ax=b; solve equations of the form ef(x) = b or disguised quadratics; BAT Use the change of base formula for logarithmsKey Q: how can we solve an equation when x is a power;

A1103WB17-21

Starter rearranging algebraic terms using laws logarithms and quick review of laws of logarithms. Can extend with more questions if appropriateWB17- 19 model taking logarithms of both sides and rearranging to solve index equations with x as a power. WB20-21 model solving disguised quadratics with index terms and x as power. Students should understand that the more steps a solution takes the more room for error. So emphasise checking answers with calculator. Use Solomon worksheets for lots practice An ability to solve equations of the form eax+b=p and ln (ax+b)=q is expected.In solving equations students may use the change of base formula. Solving equations questions may be in the form 23 x−1=3.

Key Q: how can we solve an equation when there are logarithms with different bases;

A1104WB22

Starter simplify calculations with logarithm lawsIntroduce change of base formula with a couple of historical questions that can be directly checked with a calculator – discuss the advantage of a calculator versus logarithm tables. WB22 using the change of base formula to solve equations with logarithms of different base. Short discussion and practice since unlikely in exam.

BAT solve problems with growth functions and logarithms in a modelling contextKey Q: how do we apply logarithms in real life contexts

NEEDS MORE examples including basic growth fn qsNEEDS MORE examplesNEEDS MORE examplesNEEDS MORE examples

A1105WB23

StarterWB23 models using straight line graph of log y against log x to explore relationship between number of microbes in a culture and time.Students should be able to plot log y against log x and obtain a straight line where the intercept is log a and the gradient is n and plot log y against x to obtain a straight line where the intercept is log k and the gradient is log b. There should be discussion about why this is an appropriate model and why it is only an estimate.Contexts for modelling should could include the use of e in continuous compound interest, radioactive decay, drug concentration decay, exponential growth as a model for population growth. Students should be familiar with terms such as initial, meaning when t = 0. They may need to explore the behaviour for large values of t or to consider whether the range of values predicted is appropriate. Consideration of a second improved model may be required.

BAT know that the gradient of ekx is equal to k ekx and hence understand why the exponential model is suitable in many applications; Key Q:What follows: differentiation and integration;

A108b LOGARITHMS and EXPONENTIALS6.1 Know and use the function ex and its graph6.2 the gradient of ekx is equal to k ekx and hence understand why the exponential model is suitable in many applications6.3 the function ln x and its graph; use ln x as the inverse function of ex

6.7 use exp growth and decay, use in modelling (radioactive decay, drug concentration decay, continuous compound interest); consider limitations of models


BAT the function ln x and its graph; use ln x as the inverse function of ex; solve equations with ex and ln xKey Q: can we apply what we know about logarithms and growth functions to solve equations with exp and ln x;

Practice set qs to doPractice set qs to doPractice set qs to do

A1110WB1-6

Starter solving equations using laws of logarithm and indices. Then review activity: write an example of each of the laws displayed. Check students know the laws and extend with harder examples if neededWB1-2 show same laws and explain that exp and ln x are ‘special functions’ but that the laws can be applied to them and model examples of simple equations withex and ln xterms. Building up to more complex equations. Questions requiring exact answers and rounded answers to show that either can be required, especially getting used to exact answers for proof questions. WB3-4 develop solving equations into harder problems with: applying the laws of logarithms and indices to simplify to single terms; multi-step solutions. .. followed by practice set of questions and peer questions ACTIVITY. WB5-6 exam level questions to review and fix any misconceptions.

BAT solve equations with ex and ln xKey Q: can we solve harder equations that involve exp and ln x – disguised quadratics;

Practice set qs to doPractice set qs to doPractice set qs to do

A1111WB7-8

Starter rearranging using laws of logarithms. Alternatively give students some basic equations with exp and ln x to solveWB7-8 model solving quadratic where variable term is ex or ln x. Showing extra steps for solution. WB9 demonstrates when some solutions are invalid. Followed by practice set of questions

BAT Know and use the function ex and its graph; know that the gradient of ekx is equal to k ekx and hence understand why use exp growth and decay, ; apply transformations to exp and ln x graphs Key Q: why have a special growth function and inverse; what properties do exp and ln x have;

A1112WB9-10

A1113WB11-12

Starter using geogebra explore the domain and range of given functions. Students have not met domain and range as concepts so simplify language for them to’ inputs that are possible for x and outputs that are possible for y’. WB9 explores a calculation with compound interest that leads to derivation of the exact value of e. Can then discuss e as a special value similar to PI in some ways that we apply its use. WB10 use the geogebra file for WB10 and explore the gradient of growth functions using the step by step notes pages on the PPT. Leading to an understanding of the exp function as a special growth function where the gradient equals the value of y. the summary briefly explains why the exp function is used in models of continuous growth.

WB11-12 graph sketching ACTIVITY. Can use mini-whiteboards. Go through types

of transformations applied to exp and ln graph. Mostly exp graph. WB13-14 Use mini-whiteboards for sketching transformations of natural logarithm graphs Followed by chili pepper challenge – peer questions.

BAT use in modelling (radioactive decay, drug concentration decay, continuous compound interest); consider limitations of models Key Q: Add some tangent normal geometry??? Add some tangent normal geometryAnswers to fill in with calculator wb 15-17

A1114WB15-17

Starter – discuss situations where exp functions are a suitable model – pages on PPT give examples. WB15 example of price of a used car. Substituting values to function, interpret and sketching graphWB16 modelling population elephants. Substituting, solving equation by taking logarithms; finding maximum value of function by taking limits WB17 population of farmers on island. Finding initial population; sketch graph; substitution and solve equation taking logarithmsMake sure students grasp the common features of modelling questions i.e. it’s the same application of maths throughout

BAT know that the gradient of ekx is equal to k ekx and hence understand why the exponential model is suitable in many applications; Key Q: how can we find the rate of change (gradient) n a model of growth?Add model where gradient used ?? velocity graph? Rate of change of population? Rate of deprediation of car value?Not doneNot done answer to modelling QNot doneNot done

A1115WB18-19

Proof from first principles?Starter laws of logarithms Either prove from first principles that the derivative of exp is exp or refer back to graphical work using WB10WB18 model differentiation and integration of exp (x) function and extend to exp (kx). This is only being met briefly but is expected knowledge when we meet full differentiation topic in the A2 course. WB19 demonstrates where the rate of change of a growth function is used in a model of a real life situation. Substitution to function; finding the gradient by differentiation; sketching both function and gradient function on same axes accurately.

What follows: differentiation and integration in A2 content such as chain rule; composite and inverse functions ;

hours guided learning time BAT Key Q:

PPT WB

BAT Key Q:

PPT WB

BAT Key Q:

PPT WB

BAT Key Q:BAT Key Q:

Web viewStarter puzzle with differences of two squares. ... getting students to demonstrate answers...

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Transcript of Web viewStarter puzzle with differences of two squares. ... getting students to demonstrate answers...