Mathematics units Grade 11 advanced - csomathscience · PDF fileMathematics units: Grade 11 advanced 135 teaching hours UNIT 11A.4: Trigonometry 1 Sine and cosine rule Solution of

Mathematics units Grade 11 advanced

Contents

11A.1 Number and algebra 139 11A.8 Algebra 3 199

11A.2 Geometry 1 149 11A.9 Trigonometry 2 209

11A.3 Algebra 1 157 11A.10 Probability 2 217

11A.4 Trigonometry 1 165 11A.11 Calculus 225

11A.5 Probability 1 175 11A.12 Vectors 237

11A.6 Algebra 2 185 11A.13 Geometry 2 245

11A.7 Measures 193 11A.14 Algebra 4 253

Mathematics units: Grade 11 advanced 135 teaching hours

UNIT 11A.4: Trigonometry 1Sine and cosine ruleSolution of triangles in 2-D and 3-D9 hours

UNIT 11A.10: Probability 2RiskTrends over time; movingaveragesSimulations using randomnumbers10 hours

15% 30%

1st semester70 hours

2nd semester65 hours

UNIT 11A.2: Geometry 1ProofStandard circle theorems9 hours

UNIT 11A.7: MeasuresRates and compoundmeasures3 hours

UNIT 11A.5: Probability 1Empirical probabilityUsing mathematical models,e.g. tree diagramsIndependent and dependentevents10 hours

UNIT 11A.9: Trigonometry 2Circular functionsTrigonometric equations and identities7 hours

Reasoning and problem

solving should be integrated into each unit

UNIT 11A.0: Grade 10A revision3 hours

UNIT 11A.1: Number and algebraSequences: finite and infinitegeometric sequences; sum of firstn squares and cubes; recurrencerelationsBinomial theoremPermutations and combinations12 hours

UNIT 11A.3: Algebra 1Properties of graphs of functions,including maxima and minimaQuadratic functions12 hours

UNIT 11A.8: Algebra 3Cubic, reciprocal, sine and cosinefunctionsModulus and other non-standardfunctionsInverse functionsComposite functions12 hoursUNIT 11A.11: Calculus

LimitsIntroduction to calculusDerivatives of standard functions12 hours

UNIT 11A.6: Algebra 2Quadratic equations; real rootsSimultaneous equations (linear andquadratic)Inequalities, including solution sets12 hours

UNIT 11A.14: Algebra 4Transformation of functionsExponential functionLogarithms11 hours

55%

UNIT 11A.13: Geometry 2TransformationsPlans and elevations7 hours

UNIT 11A.12: VectorsPosition vector; addition and subtractionin 2-D and 3-D; vector diagramsScalar product; multiplication by scalar;magnitude and direction; displacementand velocity; unit vectors6 hours

139 | Qatar mathematics scheme of work | Grade 11 advanced | Unit 11A.1 | Number and algebra © Education Institute 2005

GRADE 11A: Number and algebra

Series, combinatorics and the binomial theorem

About this unit This is the only unit on number and algebra for Grade 11 advanced. It brings together several topics on algebra related to sequences and combinatorics. The unit builds on work in number and algebra in the Grade 10 advanced units.

The unit is designed to guide your planning and teaching of mathematics lessons. It provides a link between the standards for mathematics and your lesson plans.

The teaching and learning activities should help you to plan the content and pace of lessons. Adapt the ideas to meet your students’ needs. Supplement the activities where necessary with appropriate tasks and exercises from textbooks and other resources, including ICT.

For consolidation or extension activities, look at the units for Grade 10 advanced or Grade 12 advanced.

Introduce the unit to students by summarising what they will learn and how this builds on earlier work. Review the unit at the end, drawing out the main learning points, links to other work and real-world applications.

Previous learning To meet the expectations of this unit, students should already be able to investigate the growth of simple patterns, and to generalise algebraic relationships to model the behaviour of the patterns. They should be able to identify and sum arithmetic and geometric series, and to convert any recurring decimal to an exact fraction.

Expectations By the end of the unit, students will work to expected degrees of accuracy. They will recognise when to use ICT and do so efficiently. They will use sigma notation, and will generate and sum simple recursive sequences, including arithmetic and geometric series, to model the behaviour of real-world situations. They will use formulae for the sum of the squares and cubes of the first n positive integers. They will be familiar with the patterns in Pascal’s triangle, find combinations and permutations, and use the binomial theorem expansion of (1 + x)n, where n is a positive integer.

Students who progress further will differentiate between different kinds of sequences and series with fluency, and manipulate formulae associated with them with confidence. They will find more complex permutations and combinations, and will extend their use of the binomial theorem.

Resources The main resources needed for this unit are: • overhead projector (OHP) • Internet access, computer and data projector • spreadsheet software such as Microsoft Excel • computers with Internet access and spreadsheet software for students • calculators for students

Key vocabulary and technical terms Students should understand, use and spell correctly: • recursive, sequence, series, convergent • arithmetic, geometric, common difference, common ratio, sum to n terms,

sum to infinity • permutation, arrangement, selection, combination, factorial • binomial theorem, binomial coefficient, Pascal’s triangle

UNIT 11A.1 12 hours


Standards for the unit

12 hours SUPPORTING STANDARDS Grade 10A standards

CORE STANDARDS Grade 11A standards

EXTENSION STANDARDS Grade 12A standards

11A.1.13 Work to expected degrees of accuracy, and know when an exact solution is appropriate.

11A.1.14 Recognise when to use ICT and when not to, and use it efficiently.

12AQ.1.16

11A.4.1 Generate sequences from term-to-term definitions and from position-to-term definitions, including recursive sequences, to model the behaviour of real-world situations, for example population growth.

Recognise when to use ICT and when not to, and use it efficiently; use ICT to present findings and conclusions. 10A.4.2 Generate sequences from term-to-

term and position-to-term definitions; investigate the growth of simple patterns, generalising algebraic relationships to model the behaviour of the patterns.

11A.4.2 Understand and use sigma notation for summing the terms of a sequence.

10A.4.3 Identify and sum arithmetic sequences, including the first n consecutive positive integers, and give a ‘geometric proof’ for the formulae for these sums.

11A.4.3 Recognise an arithmetic progression (AP); sum an arithmetic series and know the formula for the rth term of the series in terms of the first term and the common difference between terms.

10A.4.4 Identify and sum geometric sequences and know the conditions under which an infinite geometric series can be summed.

11A.4.4 Recognise a geometric progression (GP); generate term-to-term and position-to-term definitions for the terms of a GP in terms of the common ratio between terms; sum a finite geometric series.

10A.4.5 Convert any recurring decimal to an exact fraction.

11A.4.5 Sum to infinity a convergent geometric series and know the condition on the common ratio for an infinite geometric series to be convergent.

11A.4.6 Understand and use formulae for the sum of: • the squares of the first n positive integers; • the cubes of the first n positive integers.

2 hours

Series notation and formulae

2 hours

Arithmetic and geometric series

8 hours

Binomial theorem and combinatorics

11A.4.7 Understand and use factorial notation and know that 0! = 1; know the binomial theorem expansion of (1 + x)n for positive integer n and that the term in xr has coefficient nCr, where nCr =

!( )! !n

n r r−; know how to use Pascal’s triangle to find

nCr; find permutations and combinations.

12AQ.4.1 Understand that nCr is the number of combinations of r different objects from n different objects and that the number of permutations of r different objects from n different objects is r! nCr, which is denoted by nPr.

12AS.4.1 Find permutations and combinations.

Unit 11A.1


Activities

Objectives Possible teaching activities Notes School resources

2 hours

Series notation and formulae Understand and use sigma notation for summing the terms of a sequence.

Generate sequences from term-to-term definitions and from position-to-term definitions, including recursive sequences, to model the behaviour of real-world situations, for example population growth.

Understand and use formulae for the sum of: • the squares of the first n

positive integers; • the cubes of the first n

positive integers.

Series notation

Class discussion Begin by explaining sigma notation as a compact way to present the sum of a set of numbers or the terms of a sequence. Do this first by a number of simple examples, and then displayed on a spreadsheet column, where the row() function can replace the index on sigma.

Conclude the presentation with a number of examples worked orally, turning series into sigma notation with index and limits, and vice versa.

Examples • Express in full:

5

1( 1)

rr r

=

+∑

• Use sigma notation to express 7 6 5 25 4 3

+ + + .

A1=row(A1)*(row(A1)+1)

B1=2; B2=B1+A2.

A1 and B2 are replicated down the columns.

Extend discussion using the applet Sequences (www.fi.uu.nl/wisweb/welcome_en.html).

This column is for schools to note their own resources, e.g. textbooks, worksheets.

With more able pupils mention of the corresponding pi notation for products may be of interest.

Exercises Get students to use the row() function in a spreadsheet to simulate the index in the sum as shown above. Many of the textbook exercises designed for pencil and paper can be adapted to this. Extend such exercises to incorporate the formulae for the sum of the first n natural numbers, the sum of their squares, and the sum of their cubes. • Express 5 × 6 + 6 × 7 + 7 × 8 + 8 × 9 + … + 199 × 200 using sigma notation, and hence, with

the aid of a spreadsheet, evaluate their sum. • Use replication to generate the first 50 terms of the series 5, 5, 10, 15, 25, … Find its sum.

• Use replication to show the terms of 100

1 1r

rr= +∑ on a spreadsheet; hence calculate the sum.

• If 2 16

1( 1)(2 1),

n

rr n n n

=

= + +∑ find a formula for 2

2

1.

n

r nr

= +∑

• If un = n(n + 1)(n + 2), write down and simplify an expression for un+1 / un, and hence obtain a

recurrence relation for the sequence (un).

Exercises like these promote new ways of looking at this sort of problem. They can lead to applications where algebra models situations such as accumulating bank deposits, growth, appreciation and decay.

Use the applet Discrete dynamic models from www.fi.uu.nl/wisweb/welcome_en.html to explore and discuss some real-world situations.

Unit 11A.1



Standard series: arithmetic and geometric

Class discussion Arithmetic and geometric series and the standard formulae for them were introduced in Grade 10 advanced. Restate them with sigma notation to revise their use.

Ask students to explain the term arithmetic sequence with one or more examples. Draw out the notation usually used: • a for the first term; • d for the common difference; • n for the number of terms.

Revise the formulae for the nth term and the sum of n terms.

Repeat the exercise for geometric series, including the notation r for the common difference. Revise the formulae for the nth term and the sum of n terms.

In both cases, discuss the proof of the sum in as much detail as is necessary. Turn to convergence. Ask for an examples of: • an arithmetic series which converges; • a geometric series which converges. Students should be able to decide that no arithmetic series converges, and will probably recall geometric series that converge. Use the discussion to re-establish the criterion ( < 1r ) for convergence.

On the web The work on arithmetic series and geometric series is featured on MathsNet at www.mathsnet.net/asa2/2004/c2.html#1.

2 hours

Arithmetic and geometric series Recognise an arithmetic progression (AP); sum an arithmetic series and know the formula for the rth term of the series in terms of the first term and the common difference between terms.

Recognise a geometric progression (GP); generate term-to-term and position-to-term definitions for the terms of a GP in terms of the common ratio between terms; sum a finite geometric series.

Sum to infinity a convergent geometric series and know the condition on the common ratio for an infinite geometric series to be convergent. Exercises

Combine a mix of APs and GPs, ending with problem solving. Include • routine exercises on the use of the formulae

(nth term, sum to n terms, sum to infinity for a GP); • calculation of fractions equivalent to recurring decimals by considering them as convergent

geometric series; • word problems that bring in applications.

There are two different models of growth involved here. Students need experience of a range of problems so that they are clear about how these growth models compare.

If time permits at the end of this work, repeat the argument that establishes that all rational numbers correspond to terminating or repeating decimals. This reinforces work on the existence of irrationals characterised by non-repeating but non-terminating decimal expansions. Alternatively, cover these points within the exercise once the routine examples have been finished.

Include problems such as this one. • The first, second and fourth terms of a

geometric progression are also in arithmetic progression. Find the possible values of the common ratio.



Arrangements (or permutations)

Class discussion Discuss this problem. • If towns A and B are connected by r distinct routes and towns B and C by s distinct routes,

how many distinct routes are there from A to C via B?

This simple problem establishes an important principle: that in such circumstances the answer is r × s. (In the diagram the number of routes from A to C is 6 rather than 5.)

Now develop the argument for the number of permutations of n objects as n!. Develop the number of permutations of r elements selected from n from this. The first choice can be made in n ways, the second in n – 1, and so on, until the rth can be made in n – r + 1. Hence by the r × s principle above, we have:

!P ( 1)( 2)...( 1)( )!

nr

nn n n n rn r

= − − − + =−

Take this example of use of factorial notation slowly, as students need to use it too. Consider the special case of 0! = 1; do this by trying to get students to tell you what it should be so that the notation remains consistent (e.g. in how many ways can 0 be arranged from n?).

The objective of this part of the work is to embed in students’ minds several distinct problems and their standardised solutions so that the same situations can be recognised as elements of more complex problems.

8 hours

Binomial theorem and combinatorics Work to expected degrees of accuracy, and know when an exact solution is appropriate.

Recognise when to use ICT and when not to, and use it efficiently.

Understand and use factorial notation and know that 0! = 1; know the binomial theorem expansion of (1 + x)n for positive integer n and that the term in xr has coefficient nCr, where nCr = !

!( )! ;nr n r− know how

to use Pascal’s triangle to find nCr ; find permutations and combinations.

Exercise Include: • basic drill on factorial notation; • occurrence of 0! in settings which make its value of 1 clearly necessary for consistency; • basic drill on permutations; • problem solving where non-standard counting situations figure.

Examples • How many ways can different groupings of four be taken for a photograph from a family

gathering of ten people? (Take the same four in a different order as a different grouping.) If Bader and Ali are inseparable (and insist on being photographed together or not at all), in how many ways can this be done?

• In how many ways can the digits in 98 765 be arranged if an even number must appear in the first, third or fifth place?

There are many sets of examples on permutations that will support this work.

Include ways of dealing with repetitions. For example, how many ways can n objects be arranged if three of them are identical? If we take any one arrangement, regarding the three identical items as distinguishable, they can be rearranged in 3! ways; hence the total number of arrangements is n! / 3!.

Also include ways of dealing with restrictions. For example, how many ways can n objects be arranged if two of them must be separate? We count the arrangements in which the two are together, which is equivalent to the number of arrangements of n – 1 items, except that the two may be either way round, so it is 2 × (n – 1)!. These are the arrangements to be excluded, so the required figure is: n! – 2 × (n – 1)! = (n – 2) × (n – 1)!



Selections (or combinations)

Class discussion Establish the concept of selection (or combination) as distinct from permutation (or arrangement). Have available a number of examples where the distinction is important.

The formula for nPr can then be adapted to that for nCr. Since any selection of r can be rearranged in r! ways, nPr = r! × nCr. Hence

!C( )! !

nr

nn r r

=−

Use this result to discuss a number of examples, including at least one non-standard problem. Do not expect too much at this stage, however, and focus clearly on repeatedly distinguishing selections (or combinations) from arrangements (or permutations). The words do not immediately connect with intuition, so students will easily remain confused.

!P ( 1)...( 1)( )!

nr

n n n n rn r

= = − − +−

where the number of factors is equal to the number of elements of the permutation.

! ( 1)...( 1)C( )! ! 1 2 ...

nr

n n n n rn r r r

− − += =− × × ×

where the number of factors in numerator and denominator is equal.

On the web

The work in this section is featured on MathsNet (www.mathsnet.net/asa2/2004/c2.html#1).

Exercise (and subsequent discussion) There are many sets of exercises available on this topic. Include • drill on the new notation nCr; • discovery that nCr = nCn–r, both from factorials and from context; • distinguishing permutations from combinations; • problem solving which focuses on counting.

Bring out in full discussion of these questions the different strategies which can lead to an equivalent answer. More able pupils will benefit from seeing that there is indeed more than one way of getting an answer and that sometimes one method is much more elegant or economical than another.

Examples • In how many ways can a committee of 5 be chosen from 10 people (a) so as to include both

the youngest and the oldest; (b) so as to exclude the youngest if it includes the oldest? • Find n if (a) nC2 = 55, (b) nC2 = nC5. • A committee of 6 is chosen from 10 men and 7 women so as to contain at least 3 men and 2

women. In how many ways can this be done if two particular women refuse to serve on the same committee?

More able students will cope with the identity n+1Cr = nCr + nCr–1 by factorials. The exercise is tricky but demanding, and makes a good contrast with the simpler combinatorial argument which can also justify it!

This result underlies Pascal’s triangle.

On the web

There are lots of websites containing interactive versions of Pascal’s triangle which help students to find the patterns, e.g. mathforum.org/ workshops/usi/pascal/mo.pascal.html.



The binomial theorem

Class discussion Pascal’s triangle was used in Grade 10 to prepare for the binomial theorem. Review the Grade 10 work briefly.

Students will recognise that the expansion of (a + b)n can be done by using the nth row of the triangle, and will remember (if hazily) how each line can be derived from the previous line. Include in the review the long multiplication method of generating each new power from the last to make clear why the rule of adding two terms from above to obtain the one below works.

Pose the problem of enumerating the nth line of the triangle without writing out all the lines that precede it. Give students time to discuss this problem in small groups. This is an extension of the use of formulae for nth terms or for the sum of n terms, which students have already met.

There are two ways of going about this. Each has its merits. Present both methods to students but in different lessons.

The simpler argument is direct appeal to combinatorics. Ask students: • How does the a5b7 term arise in the expansion of (a + b)12?

The question should provoke thought. With some encouragement it should become clear that the problem is like the routes between towns at the beginning of the unit. There are 212 routes through the brackets since there is a choice of two (a or b) each time. The term in question arises from the number of routes through the brackets which pass through exactly five of the letters a, so that the rest of the letters are b; that is 12C5.

Make the proposition based on the argument just considered that the rth term on the nth row is nCr, remembering that there is a ‘row zero’ and a term numbered zero. The argument has in effect proved that proposition. Get students to verify this on any of the early rows.

Postpone discussion of the second method (which uses a recursion relation to generate one row from the next). Move on to the use of the new formula. Derive, for example, the first four terms of (a + b)12 to show that this can now be done without Pascal’s triangle.

The applet Pascal’s triangle is a useful visual aid (see nlvm.usu.edu/en/nav/vlibrary.html).

This illustrates one possible ‘route’ through the brackets to obtain a term a5b7.

Class exercise (with problem solving) Drill on the binomial theorem is part of Grade 10; it is appropriate to revise it here but also to press on to consider harder questions with larger values of the index n and to investigate some of the properties of Pascal’s triangle in more sophisticated terms.

Include: • elementary expansions of (a + b)n in full; • examples of the type (1 + x + x2)4 where the first few terms are required; • problems on the structure of Pascal’s triangle, such as: – show that the coefficient of x in (1 + x)n is the nth triangular number;

– prove that 0

( 1) 0n

r

r

nr=

⎛ ⎞− =⎜ ⎟

⎝ ⎠∑ .



Harder problems with the binomial theorem

Class discussion Ask questions about a possible trinomial theorem. For example: • How would you expand (x + y + z)n?

Allow time for the suggestion of treating the problem as (x + [y + z])n to emerge, i.e. to use a repeated application of the binomial theorem.

Work an example such as finding the first four terms of the expansion of (1 + x + 2x2)5 (see the model solution on the right).

Make sure that students appreciate not only the strategy for solving that kind of problem but also the need to organise their work so that accuracy does not become a problem.

Model solution 2 5

2 2 2

2 3

2 3

3 2 2

2 3 3

2 3

(1 2 )5 5 41 ( 2 ) ( 2 )1 1 25 4 3 ( 2 ) ...1 2 3

1 5 10 10( 4 ...)310( 2 ...)...1

1 5 10 10 40 10 ...1 15 10 50 ...

x x

x x x x

x x

x x x x

x x x

x x x x xx x x

+ +⋅= + + + +⋅

⋅ ⋅+ + +⋅ ⋅

= + + + + +

+ + ⋅ ⋅ +

= + + + + + += + + + +

Work through some harder applications, such as this one. • Find the exact values of the sum and the product of + 7(2 3) and − 7(2 3) . Hence show that the integral part of + 7(2 3) is 10 083.

Some non-routine problems on the binomial theorem can be found by choosing the Maths Finder option on the Nrich website at www.nrich.maths.org/public/index.php.

Model solution 7

7 6 5

4 3 2

2 2 3 3

7 5 3 2

3 6 4

2 2 3

7

7 7

7

(2 3)

2 7 2 3 21 2 3

35 2 3 3 35 2 3

21 2 3 3 7 2 3 3 32 21 2 3 35 2 3

7 2 3 7 2 3 35 2 3 3

21 2 3 3 3 3

5042 2911 3

Similarly (2 3) 5042 2911 3

Hence (2 3) (2 3) 10084

Now (2 3)

+

= + ⋅ ⋅ + ⋅ ⋅

+ ⋅ ⋅ ⋅ + ⋅ ⋅

+ ⋅ ⋅ ⋅ + ⋅ ⋅ + ⋅= + ⋅ ⋅ + ⋅ ⋅

+ ⋅ ⋅ + ⋅ ⋅ + ⋅ ⋅ ⋅

+ ⋅ ⋅ ⋅ + ⋅

= +

− = −

+ + − =

+ ×27 2 7

77

7

(2 3) (2 3 ) 1Then we have

10 (2 3) 1 (2 3)

So 0 10084 (2 3) 1and the result follows.

− = − =

< − = <+

< − − <


Assessment

Examples of assessment tasks and questions Notes School resources

In a TV quiz show a contestant can triple her winnings if she survives from one round to the next. Write down a formula for her prize Pn+1 in the (n + 1)th round in terms of her prize Pn in the nth round. Write an alternative formula for Pn+1 in terms of n.

The prize for winning in the first round is QR 1000. What is the minimum number of rounds that will have to be contested to win at least QR 700 000?

A sequence is defined by un+2 = un+1 – un with u1 = 5 and u2 = –4. Write down the first eight terms of the sequence.

A sequence is defined by un = n(n – 1) + 41. Write down the first twelve terms of this sequence. What do you notice about these terms? Form a conjecture about this sequence and carry out further tests to see if your conjecture is correct.

Assessment Set up activities that allow students to demonstrate what they have learned in this unit. The activities can be provided informally or formally during and at the end of the unit, or for homework. They can be selected from the teaching activities or can be new experiences. Choose tasks and questions from the examples on the right to incorporate in the activities.

In a certain country, there is a net increase in population from one year to the next of 5%. Set up a recurrence relation to describe the population in year n + 1 in terms of the population in year n. Find the population in year n + 4 compared to the population in year n. Use your formula to find the number of years it takes to double the population from year n.

A woman buys a car and pays in monthly instalments. The car costs QR 60 500 and interest is charged on any outstanding debt at a monthly rate of r%. The woman pays back a fixed amount each month of QR M. Set up a recurrence relation connecting the amount owed, An+1, after n + 1 months in terms of the amount owed, An, at the end of the nth month. How many months will it take to repay the debt if M = 1200 and r = 1.2? How much will the woman have then paid for the car? Investigate repayments for different values of M and r.

Find 10

1( 1).r +∑

Find 2r∑ for integer values of r from 1 to 10.

Write out in full

4

1

( 1)( 1)

r

r r−

+∑ and use partial fractions to evaluate this sum.

Rewrite these sums using sigma notation:

72 + 82 + 92 + 102 1⁄121 – 1⁄144 + 1⁄169 – 1⁄196 + 1⁄225

At the end of every year a car loses 30% of its value at the start of the year. Construct a formula, in terms of the original purchase price, to give the value of the car n years after purchase. After how many years will the car first be worth less than 90% of its original value?

Show that ( 1)( 2)...( 1)C .

!n

rn n n n r

r− − − +=

Unit 11A.1



Use the binomial theorem to expand (x – 2y)4.

Fifteen European countries are about to set up three new regulatory bodies, the Avocado Authority, the Broccoli Board and the Courgette Commission. Each of the new bodies to be set up will have one member from each of 6 different countries. Some countries may be represented on more than one body, and some may be represented on none.

a. In how many ways can the 6 countries represented on the Avocado Authority be chosen?

Find the number of ways in which the three bodies can be made up in each of the following cases. Give your answers in scientific notation to three significant figures (e.g. 1.23 × 104). Note that the order in which countries are allocated to bodies does not matter.

b. The 6 countries for each body are chosen freely from the 15 original countries.

c. France insists on being represented on each body and Britain insists on not being represented on any of them, but otherwise the countries are chosen freely.

d. No country may be on both the Avocado Authority and the Broccoli Board, but otherwise the countries are chosen freely.

MEI

In this question, a circle consists of a sequence of sectors with angles a1, a2, a3, …, as shown in the figure. All angles are measured in degrees. Four cases are considered.

a. In the first case, the angles a1, a2, a3, a4, a5, a6, … form a periodic sequence 1°, 2°, 3°, 1°, 2°, 3°, … How many sectors will fill the circle?

b. In the second case, a1 = 8.5° and the angles form an arithmetic progression with common difference 1°. Verify that 20 sectors fill the circle exactly.

c. In the third case, the angles form an arithmetic progression with common difference 0.5°, and 30 sectors fill the circle exactly. Find a1.

d. In the fourth case, the angles form a geometric progression with a1 = 90° and common ratio 3⁄4. Find how many sectors have angle greater than 1°. Now show that no matter how many sectors are used they will always fit into the circle.

MEI

149 | Qatar mathematics scheme of work | Grade 11 advanced | Unit 11A.2 | Geometry 1 © Supreme Education Council 2005

GRADE 11A: Geometry 1

Circles and proofs

About this unit This is the first of two units on geometry for Grade 11 advanced. It introduces the circle theorems, building on the approach started in Grade 10.





Previous learning To meet the expectations of this unit, students should already be able to prove that the perpendicular from the centre of a circle to a chord bisects the chord; and that the two tangents from an external point to a circle are of equal length.

ExpectationBy the end of the unit, students will use geometry to solve theoretical problems. They will prove standard circle theorems. They will use a dynamic geometry system to conjecture results and to explore geometric proof.

Students who progress further will solve more complex geometrical problems.

Resources The main resources needed for this unit are: • overhead projector (OHP) • Internet access, computer and data projector • dynamic geometry system (DGS) such as:

Geometer’s Sketchpad (see www.keypress.com/sketchpad) Cabri Geometrie (see www.chartwellyorke.com/cabri.html)

• computers with Internet access and dynamic geometry software for students

• graphics calculators for students

Key vocabulary and technical terms Students should understand, use and spell correctly: • proof, theorem, converse • angle subtended by an arc at the centre and at the circumference • major arc, minor arc, segment, sector, tangent, chord, cyclic quadrilateral,

concyclic, collinear

UNIT 11A.2 9 hours






10A.1.6 Develop short chains of logical reasoning, using correct mathematical notation and terms.

11A.1.6 Develop chains of logical reasoning, using correct mathematical notation and terms.

12AS.1.6

11A.1.7 Explain their reasoning, both orally and in writing.

10A.1.8 Generate simple mathematical proofs, and identify exceptional cases.

11A.1.8 Generate mathematical proofs, and identify exceptional cases.

Develop chains of logical reasoning, using correct terminology and mathematical notation, including symbols for logical implication.

10A.1.9 Generalise whenever possible. 11A.1.11 Conjecture alternative possibilities with ‘What if …?’ and ‘What if not …?’ questions.



12AQ.1.16

11A.8.1 Use a dynamic geometry system to conjecture results and to explore geometric proof.

Recognise when to use ICT and when not to, and use it efficiently; use ICT to present findings and conclusions.

2 hours

Circle theorems 1

2 hours

Circle theorems 2

3 hours

Circle theorems 3

2 hours

Proofs

10A.6.9 Prove the circle theorems: • The perpendicular from the centre

of the circle to a chord bisects the chord.

• The two tangents from an external point to a circle are of equal length.

11A.8.9 Prove the circle theorems: • The angle subtended by an arc at the centre of the circle is

twice the angle subtended by the arc at a point on the circle, including, as a special case, the angle in a semicircle is a right angle.

• Angles in the same segment subtended by a chord are equal. • The angle subtended by a chord at the centre of a circle is

twice the angle between the chord and the tangent to the circle at an end point of the chord.

• When two chords BC and DE in a circle intersect at A then AB × AC = AD × DE.

• Opposite angles of a cyclic quadrilateral are supplementary.

Unit 11A.2


Activities


Investigating circle properties

Investigation Begin by asking students to use a dynamic geometry system to investigate the relationship between the angle at the centre of a circle subtended by an arc and an angle subtended by the same arc at the circumference. Structure this as follows. • Draw a circle and mark two points A and B on it that are not opposite ends of a diameter. • Draw the two radii joining the points to the centre of the circle O. • Measure angle AOB and label the angle with that measurement. Check that your

measurement changes dynamically as you move A or B on the circumference of the circle. • What is the measured angle when A and B are at opposite ends of a diameter? Are there any

other special cases? • With points A and B more or less in their original positions, mark a third point C, distinct from

both A and B, on the major arc AB. • Measure and label angle ACB. • What do you notice if you move A or B as before? Can you turn that observation into a

conjecture? Is it restricted in any way? Are there any special cases? • Check that your conjecture is still valid if this time you move C. Is there any restriction on C? • Can you prove your conjecture?

This figure was produced in Cabri Geometrie. It shows the configuration used in the investigation. On the web Waldo’s Interactive Maths pages have sections on the circle theorems. See www.waldomaths.com.


2 hours

Circle theorems 1 Develop chains of logical reasoning, using correct mathematical notation and terms.

Explain their reasoning, both orally and in writing.

Conjecture alternative possibilities with ‘What if …?’ and ‘What if not …?’ questions.


Use a dynamic geometry system to conjecture results and to explore geometric proof.

Prove the circle theorems: • The angle subtended by an

arc at the centre of the circle is twice the angle subtended by the arc at a point on the circle, including, as a special case, the angle in semicircle is a right angle.

• Angles in the same segment subtended by a chord are equal.

Class discussion Help students to draw together their conclusions. Not all their conclusions will be the same or expressed in the same way. Most students should realise that the angle at the circumference is always half the angle at the centre (once they have set aside the way in which the result changes when C passes over A or B, or vice versa). The result that opposite angles of a cyclic quadrilateral add to 360° may emerge.

Once conclusions are settled, raise the question of proof. There are two issues: • that the results need to be proved; • how you do it.

Review the properties of an isosceles triangle, which underpin the proof (see the details on the right).

Do not rush to the hierarchy of results, namely that for a given arc: • the angle at the centre is twice the angle at the circumference; • all angles at the circumference are equal.

Rather, allow students to see first that these two results are connected: one leads to the other. This may help students to focus on the first result as the more fundamental and challenging.

To establish the result: • draw the circle and points A, B, C and O, with

lines AO, BO, AC and BC; • invite suggestions on how to proceed, then

draw OC; • establish that the two triangles are isosceles

(because of equal radii); • invite other suggestions on how to proceed; • extend CO and establish the 2x and 2y; • show that ∠AOB = 2(x + y) = 2 ∠ACB.

Unit 11A.2



Exercises Follow this with examples of calculations based on these results. A variety of simple configurations are needed which help students to see the basic configuration in different orientations.

Examples Find the angles marked a, b and c in the figures on the right.

2 hours



Generate mathematical proofs, and identify exceptional cases.


Prove the circle theorem: • Opposite angles of a cyclic

quadrilateral are supplementary.

More circle theorems

Class discussion Establish the further results about cyclic quadrilaterals, namely: • opposite angles of a cyclic quadrilateral are supplementary; • the external angle of a cyclic quadrilateral is equal to the interior opposite angle.

Do this by posing the problem on the right, with 110°, a and b marked on the diagram. Ask students to suggest ways of calculating the other two angles of the quadrilateral. Some students should suggest connecting the diagram with what has gone before by drawing in radii. It is important to do this so that students can ask ‘What if …?’ questions for themselves, and see that a certain degree of lateral thinking is essential.

Once one or two examples have been studied, allow students time to make the conjectures (the theorems just quoted) and to write out or contribute the steps of a general proof.

For reference these results can be referred to by: • opposite ∠s of a cyclic quadrilateral; • external ∠ of a cyclic quadrilateral.

These can be abbreviated if appropriate.

Steps in the calculation: • do not draw the dotted radii in first – just mark

110°, a and b; • invite suggestions on how to proceed, then

draw the dotted radii; • use the angle properties to calculate 220°,

2a (= 360° – 220°), then a, then b (= 110°).



Exercises These should extend the work in the last set of exercises. Stick to calculations for the most part.

Include the interesting result (see the diagram on the right) that produces two parallel lines AB and DC. Ask students: • If ∠BAD = 80°, what is ∠ADC? • What can you say about AB and DC in this case? • Does this result generalise?

On the web The Geometry section of MathsNet has a section on circle theorems. See www.mathsnet.net/geometry.

3 hours





Prove the circle theorems: • The angle subtended by a

chord at the centre of a circle is twice the angle between the chord and the tangent to the circle at an end point of the chord.

• When two chords BC and DE in a circle intersect at A then AB × AC = AD × DE.

The alternate segment

Class discussion Establish the tangent results. Use class discussion to establish the proof or – if appropriate for the group – use a structured investigation using a dynamic geometry system to obtain the tangent result. There are two associated results: • the alternate segment property; • the equivalent relationship specified in the standards.

The full argument is shown on the right.

In the diagram, 2x3 = 2x4 since both subtend the same chord.

x1 + y1 = 90° (∠ between tangent and radius)

y1 = y2 (isosceles )

so 2x3 = 180° – y1 – y2 = 2x1 (∠ at the centre)

and x1 = x4 (alternate segment result)



Exercises Set questions (mainly calculations) on the new properties. As this is the third extension of the work, students by now should be quick to absorb the new properties and find it much easier to make progress. More able students will benefit from trying some proof questions once they have mastered the basic drills.

Examples • In the figure on the right, calculate all the angles marked by a letter when: – e = 72°, u = 64°; – x = 58°, f = 53°. • T is a point outside a circle, and TA and TC are tangents which meet the circle at A and C

respectively. B is a point on the minor arc AC and D is a point on the major arc AC. Sketch the figure and work out as many angles as you can when:

– ∠TAB = 41°, ∠BAC = 28°, ∠CAD = 38°; – ∠ABC = 152°, ∠TCB = 10°, ∠DAC = 31°.

Intersecting chords

Class discussion Establish the intersecting chords property.

To prove the intersecting chords theorem, return to the diagram used for showing that two angles on the same arc are equal.

Refer students to the diagram, and ask them: • Which triangles are similar and why? • What does this say about the relationship between x, y, z and u? • Deduce that x × y = u × z. There is an equivalent proof when the point where the two chords intersect lies outside the circle. Students may be able to establish the first result by themselves using the hints above; if instead you do this as class discussion, get them to do the second case for themselves.

Exercises Set calculations on the new properties. This section will make a refreshing change since all the previous work has focused on angles. Use the opportunity to revise work on similarity which has been implicated in the proofs.

Examples • With respect to the figure used above right: – if x = 6 cm, y = 4 cm, u = 2 cm, calculate z; – if x + y = 10 cm, y = 7 cm, u = 3 cm, calculate u + z.

When P lies outside the circle, the cyclic quadrilateral results establish that PDE and PAB are similar. It follows that PE × PA = PD × PB. If the limit is taken as B and D approach C, then both products are also equal to PC2.



Mathematical logic

Class discussion Students have rehearsed thoroughly in stages the various angle properties associated with a circle. In addition, in the intersecting chords theorem they have revisited work on similar triangles. The work has been based on the properties of an isosceles triangle.

Take the opportunity to review the angles work by asking students to rehearse once again the arguments that establish the angle properties associated with a circle. List the various results as the discussion develops. Do one or more examples of proofs if appropriate.

Exercises These should now focus on proof. Revise if necessary the way a proof should be presented, and insist that this procedure is followed. This work will seem much more difficult so allow time for it to be mastered.

Examples • XLY is a tangent to a circle of which LM is chord. The bisectors of ∠XLM and ∠YLM cut the

circle at A and B. Prove that AB is a diameter. What sort of figure is ALBM? • Two circles touch internally at V. A line through V cuts the circles again at U and W. Prove

that the tangents at U and W are parallel. Does a similar result follow if the circles touch externally at V?

• A and C are two distinct points on a circle, not at opposite ends of a diameter. B is any point on the major arc AC. Line BD is drawn so that it bisects angle ABC and meets the circle at D. Show that, no matter where B is, point D is always the same.

• Two chords of a circle of length 10 cm and 12 cm intersect so that

the longer bisects the shorter. Find in what proportions the longer is divided by the shorter, giving your answer to two significant figures.

2 hours

Proofs Develop chains of logical reasoning, using correct mathematical notation and terms.



Generate mathematical proofs.

Class discussion To round off this unit, consider some ‘What if …?’ questions. For example, reverse the cyclic quadrilateral result. Ask: • If non-collinear points P, Q, R and S are such that ∠PQS = ∠PRS, do the points lie on a

circle?

This is similar to the converse of Pythagoras’ theorem in that it is one of the few cases of a theorem and its converse where the two are not trivially related.

Make sure that all students realise that this is a serious question that needs logical argument and proof.

On the web There is a page devoted to the intersecting chords theorem at www.cut-the-knot.org/ proofs/IntersectingChordsTheorem.shtml.

Model solution of problem in final class discussion

We start with x1 and x2 equal.

We assume that R lies outside the unique circle passing through P, Q and S.

Draw in PR′ to get a further equal angle x3.

Then PR and PR′ have corresponding angles x2 and x3 equal so must be parallel. But they intersect at P, which is a contradiction.

The same follows if we assume that R lies inside the circle. So R must lie on the circle and the four points are indeed concyclic.


Assessment


In triangle ABC, the altitudes BN and CM of the triangle ABC intersect at S. ∠ MSB is 40° and ∠ SBC is 20°. Prove that triangle ABC is an isosceles triangle.

TIMSS Grade 12

Each side of the regular hexagon ABCDEF is 10 cm long. Find the length of the diagonal AC.

TIMSS Grade 12

Assessment Set up activities that allow students to demonstrate what they have learned in this unit. The activities can be provided informally or formally during and at the end of the unit, or for homework. They can be selected from the teaching activities or can be new experiences. Choose tasks and questions from the examples on the right to incorporate in the activities. Two circles with centres at A and B have radii of 7 cm and 10 cm as shown in the

diagram. The length of the common chord PQ is 8 cm.

Calculate the length of AB.

TIMSS Grade 12

A straight line intersects a circle at points A and B. The circle in turn intersects another circle at D and C. AD is produced until it intersects the second circle at E, and BC similarly at F. Prove that AB is parallel to EF.

The diagram shows two circles, centres A and B. OPX and OQY are tangents to the circles.

a. State the mathematical name for the quadrilateral ABYQ. Give your reasons.

b. Angle AOQ is 20°. The line OAB cuts the larger circle at Z, as shown. Calculate the size of angle BZY. Show your working.

MEI

Unit 11A.2

157 | Qatar mathematics scheme of work | Grade 11 advanced | Unit 11A.3 | Algebra 1 © Education Institute 2005

GRADE 11A: Algebra 1

Quadratics and graphs

About this unit This is the first of four units on algebra for Grade 11 advanced. In it students will connect their algebraic and graphical knowledge as they see how a quadratic form relates to its graph. The unit builds on the work on algebra in the Grade 10 advanced units.





Previous learning To meet the expectations of this unit, students should already be able to solve quadratic equations exactly, by factorisation, by completing the square and by using the quadratic formula.

Expectations By the end of the unit, students will use their knowledge of interconnections in mathematics and a range of strategies to solve problems. They will find the axis of symmetry of the graph of a quadratic function, and the coordinates of its turning point. They will know what the discriminant tells them about the real roots of a quadratic equation, and they will find approximate solutions of quadratic equations using graphical methods. Through their study of functions and their graphs, and the solution of associated equations, students will appreciate a range of numerical and algebraic applications in the real world.

Students who progress further will solve more complex problems and will know what the discriminant tells them about the complex roots of a quadratic equation.

Resources The main resources needed for this unit are: • overhead projector (OHP) • Internet access, computer and data projector • spreadsheet software such as Microsoft Excel • graph plotting software such as:

Autograph (see www.autograph-math.com) Graphmatica (free from www8.pair.com/ksoft)

• computers with Internet access, spreadsheet and graph plotting software for students


Key vocabulary and technical terms Students should understand, use and spell correctly: • quadratic, linear, root, repeated • symmetry, axis, turning point, maximum, minimum • discriminant, non-negative

UNIT 11A.3 12 hours






11A.1.3 Identify and use interconnections between mathematical topics.

11A.1.5 Use a range of strategies to solve problems, including working the problem backwards and then redirecting the logic forwards; set up and solve relevant equations and perform appropriate calculations and manipulations; change the viewpoint or mathematical representation, and introduce numerical, algebraic, graphical, geometrical or statistical reasoning as necessary.

11A.5.2 Model a range of situations with appropriate quadratic functions.

11A.5.3 Find the axis of symmetry of the graph of a quadratic function, and the coordinates of its turning point by algebraic manipulation; understand the effect of varying the coefficients a, b and c in the expression ax2 + bx + c.

10A.4.11 Solve quadratic equations exactly, by factorisation, by completing the square and by using the quadratic formula.

11A.5.4 Given a quadratic equation of the form ax2 + bx + c = 0, know that: • the discriminant Δ = b2 – 4ac must be non-negative for the

exact solution set in to exist; • there are two distinct roots if Δ is positive and one repeated

root if Δ is zero.

12AQ.6.1 Given a quadratic equation of the form ax2 + bx + c = 0, know that if the discriminant Δ = b2 – 4ac is negative, there are two complex roots, which are conjugate to each other.

3 hours

Plotting and using quadratic graphs

3 hours

Completing the square

3 hours

Constructing quadratic graphs

3 hours

Modelling and problem solving with quadratics

11A.5.5 Find approximate solutions of the quadratic equation ax2 + bx + c = 0 by reading from the graph of y = ax2 + bx + c the x-coordinate(s) of the intersection point(s) of the graph of this function and the x-axis.

Unit 11A.3


Activities


3 hours

Plotting and using quadratic graphs Find approximate solutions of the quadratic equation ax2 + bx + c = 0 by reading from the graph of y = ax2 + bx + c the x-coordinate(s) of the intersection point(s) of the graph of this function and the x-axis.

Use a range of strategies to solve problems, including working the problem backwards and then redirecting the logic forwards; set up and solve relevant equations and perform appropriate calculations and manipulations; change the viewpoint or mathematical representation, and introduce numerical, algebraic, graphical, geometrical or statistical reasoning as necessary.

Plotting quadratic graphs

Class discussion Grade 10 advanced work has familiarised students with quadratic graphs using ICT. The purpose of the first section of this unit is for students to become familiar with pencil and paper techniques for graphs so that they better appreciate the advantages of using ICT.

Begin by discussing the technique for plotting curves as opposed to straight lines. Stress: • constructing a table of values; • choosing scales; • plotting points; • drawing a smooth curve.

The table of values Usually the problem will dictate the domain for x (the range of the x-scale). Where this is not the case, students must decide by experiment what range to use. Make the (usually integral) values in the domain the first row of the table, and thereafter give each term (with its sign) another row.

The range for y and the scales on the axes Aim to use scales that are a compromise between having a square-shaped graph and manageability. Avoid, for example, 3 units per 2 cm, which would be difficult both to plot and to read; and avoid tall thin or short fat graphs. Avoid also large scales that result in widely separated points; the result is an unreliable graph between the points. This advice is best offered after students have experimented a little for themselves and have put together the resultant graphs for general criticism.

Plotting points Use dots for points, so that a good curve conceals them effectively. In graphs from scientific experiments, students may have used crosses to plot points. In a mathematical graph the points are only means to an end and not in themselves of significance; nevertheless, the resulting curve must pass through all of them. In a science experiment, the points are scientific measurements that are individually important; often the aim is to get a straight line and not all the measured points will necessarily lie on it.

Joining the points Take time to allow students to practise this skill. Get students to turn the paper round to use the natural facility of the wrist to best advantage. The curve should be lightly drawn with a sharp but soft pencil first, and then firmed up as its precise trace becomes clear. Once drawn, it should not be possible to discern where the original points were by bumps of sudden curvature. Once the plots have been made, if there are wide gaps between any pairs of adjacent points, the points should be supplemented by extra ones.

Table of values for the graph y = x2 – 3x + 1 on the interval –1 ≤ x ≤ 4:

x –1 0 1 2 3 4

x2 1 0 1 4 9 16

–3x 3 0 –3 –6 –9 –12

+1 1 1 1 1 1 1

y 5 1 –1 –1 1 5

Note the distinction of the first row from the rest by heavy underlining! A frequent mistake is to add this row into the sum of each column.

This example suggests a scale of 2 cm per unit on the x-axis (so a width of 10 cm) and (in this case) the same for the y-axis (so a height of about 12 cm); the result 10 cm × 12 cm is roughly square.

Use Autograph in support of this section to show the correct answers for the graphs. Microsoft Excel can also be used to display tables such as that above quickly for the checking of answers.


Unit 11A.3



Exercises Set questions such as those on the right. Some new items of vocabulary are introduced here. At the end of this section review them for future reference. Give students enough practice to ensure that they master: • the graph plotting skills; • the correspondence between graphs and algebraic problems such as solving a quadratic.

Make sure that the questions set allow discussion of potential pitfalls, such as the distinction between 2x2 and (2x)2 and between (–x)2 and –x2.

Include at least one case of a graph that does not cut the x-axis so that its corresponding quadratic equation is insoluble.

Examples • Plot points and draw a smooth curve for the

function y = x2 + 3x – 2 for values of x which satisfy –5 ≤ x ≤ 2. For what values of x is y = 0? What are the coordinates of the turning point? Is it a maximum or a minimum?

• Draw the graph of y = 2x2 – 5x – 2 for values of x which satisfy –1 ≤ x ≤ 6. Find:

– the values of x for which y = 0; – the coordinates of the turning point; – the equation of the axis of symmetry.

How would you use the graph you have drawn to solve also the equation 0 = 2x2 – 5x – 4?

3 hours

Completing the square Identify and use interconnections between mathematical topics.

Find the axis of symmetry of the graph of a quadratic function, and the coordinates of its turning point by algebraic manipulation; understand the effect of varying the coefficients a, b and c in the expression ax2 + bx + c.

Given a quadratic equation of the form ax2 + bx + c = 0, know that: • the discriminant Δ = b2 – 4ac must be non-negative for the exact solution set in to exist;

• there are two distinct roots if Δ is positive and one repeated root if Δ is zero.

Completing the square

Class discussion

The last section introduced the idea of finding maximum and minimum values of a function. Move on to relating those values to the parameters of ax2 + bx + c, and the technique of completing the square.

Consider this question. • What is the minimum value of (say) x2 + 3x – 2?

Some students may suggest drawing a graph. Since that is time-consuming, what follows is an alternative approach.

Remind students about work in Grade 10 on solving quadratics by completing the square.

Introduce the completion of the square technique by writing: x2 + 3x – 2 = (x + 3⁄2)2 – (3⁄2)2 – 2

= (x + 3⁄2)2 – 4 1⁄4

Once the details of the technique are clear, ask students to volunteer: • the coordinates of the minimum (or maximum) point of the corresponding curve; • the equation of the axis of symmetry.

Extend the technique to cope with cases where the coefficient of x is either or both: • non-zero; • negative.

If students display little confidence in the algebra featured here, use Autograph to show that the original and the completed square form have the same graphs. On the web Quadratics, functions and equations are extensively covered on MathsNet at www.mathsnet.net/algebra.

Mathworld has a lengthy article beginning at high-school level and going beyond at mathworld.wolfram.com/QuadraticEquation.html.

St Andrews’ Mathematics has a history article at www-groups.dcs.st-and.ac.uk/~history/ HistTopics/Quadratic_etc_equations.html.



Exercises Set questions to practise this new technique which tie this work to the previous section on graphs. If students have retained their graphs, for example, they can use algebraic techniques to verify their answers or to make them more precise.

Examples • Find a and b such that x2 – 4x + 1 = (x – a)2 + b. • Find a, b and c such that 2x2 – 4x + 1 = a(x – b)2 + c. • Find the coordinates of the turning point and the axis of symmetry of the curve

y = 5 + 4x – 2x2. • Does the equation 1 + x + x2 = 0 have any solutions? Explain your answer carefully.

Using completing the square

Class discussion Conclude this section of work by considering the general equation ax2 + bx + c = 0 (a ≠ 0) and derive the quadratic formula. Emphasise that: • the formula method is based on completion of the square; • it explains how to decide which cases have roots and which do not (corresponding to curves

which do or do not cut the x-axis) according to the values of Δ = b2 – 4ac.

The full derivation of this result is included in work for Grade 10 advanced.

The three cases are • Δ > 0: two distinct roots; • Δ = 0: two coincident roots (so just one); • Δ < 0: no roots.

Exercises Set questions that require calculation of the discriminant in different contexts.

Examples • Show that the equation x2 – x + 1 = 0 has no roots. • Show that the line y = 2x – 1 does not intersect the curve y = x2 + x + 1. • Find the range of values of m for which x2 – mx + 4 = 0 has roots.



Linking algebra and graphs

Class discussion Students’ experience now encompasses a wide spectrum of techniques in relation to quadratic functions and equations.

Summarise in as much detail as needed the technique of completing the square and how the parameters that figure in the derivation relate to the representation of a function on a graph. The particular points to mention are that: • a quadratic graph is always a parabola; • the intercept on the y-axis is at (0, c); • the value of a determines the orientation as well as the ‘thinness’ of the parabola;

• the axis of symmetry is 2bxa

= − ;

• the minimum/maximum point is 24,

2 4b ac ba a

⎛ ⎞−−⎜ ⎟⎝ ⎠

;

• the corresponding quadratic equation will or will not have roots depending on the value of b2 – 4ac.

More able students may appreciate a brief digression into curves such as x = y2.

3 hours

Constructing quadratic graphs Use a range of strategies to solve problems, including working the problem backwards and then redirecting the logic forwards; set up and solve relevant equations and perform appropriate calculations and manipulations; change the viewpoint or mathematical representation, and introduce numerical, algebraic, graphical, geometrical or statistical reasoning as necessary.

Find the axis of symmetry of the graph of a quadratic function, and the coordinates of its turning point by algebraic manipulation; understand the effect of varying the coefficients a, b and c in the expression ax2 + bx + c.

Exercises

Set questions to be explored with graphics calculators (or Autograph) so that the correspondence between the algebra and graphs is exercised.

As a simple alternative to Autograph or graphics calculators, use the applet Grapher from nlvm.usu.edu/en/nav/vlibrary.html. The trace function is a particularly good feature of this applet as it allows students to see the values of x and y at particular points of the graph.

Examples • Find an equation that models the graph on the right. • Find an equation which models the graph obtained by reflecting the one shown about the line

y = –2. • Find a quadratic function with minimum value 2 and which cuts the y-axis at y = 2. • Consider the graph of y = x2. By drawing a suitable straight line, find an approximate solution

to the equation 10x2 – 2x + 25 = 0.



3 hours

Modelling and problem solving with quadratics Use a range of strategies to solve problems, including working the problem backwards and then redirecting the logic forwards; set up and solve relevant equations and perform appropriate calculations and manipulations; change the viewpoint or mathematical representation, and introduce numerical, algebraic, graphical, geometrical or statistical reasoning as necessary.

Model a range of situations with appropriate quadratic functions.

Modelling and problem solving

Class discussion Introduce a problem that is sufficiently complex to preclude guesswork and that requires a quadratic equation for solution. For this purpose, review the approach, namely: • specifying the unknown carefully, either by defining it in words or by drawing an appropriate

diagram; • formulating an equation; • solving the equation; • interpreting the solution by giving a clear answer to the original question.

Emphasise the elements of modelling implicit in this process: • attempting to represent the problem as a piece of mathematics; • solving the mathematical problem in its own terms; • relating the solution critically to the original situation.

This is a good opportunity to revise some Grade 10 work, such as factorising and solving quadratics by formula.

Exercises Set problems such as the examples below which: • require modelling techniques; • revisit work covered in Grade 10.

Examples, with model solutions on the right 1 The sum of the squares from n + 1 to n + 4 is 294. What is n?

(Hint: use the formula for the sum of squares.) 2 A herdsman took a certain number of goats to market and sold them for QR 198 in total. If he

had sold them for QR 2 less each he would have needed to sell four more to have the same takings. How many goats did he sell?

Further problem solving Some problems on quadratic functions can be found by choosing the Maths Finder option on the Nrich website (www.nrich.maths.org/public/index.php). Try the problems Parabolic patterns and More parabolic patterns.

Model solutions

Problem 1 16

16

2

2

3 2 2

3 2

2

2

( 4)( 5)(2 8 1)( 1)(2 1) 294

( 9 20)(2 9)(2 3 1) 1764

2 18 40 9 81180 2 3 176424 120 1584 0

5 66 0( 11)( 6) 0

6 or 11

n n nn n n

n n nn n n

n n n n nn n n

n nn n

n nn

+ + + +− + + =

+ + +− + + =

+ + + ++ − − − =

+ − =+ − =

+ − == −

So, given that n must be positive, it is 6.

Problem 2

Number of goats

Price per goat

n 198 / n

n + 4 198 / (n + 4)

2

2

2

198 198 24

198( 4) 198 2 ( 4)792 2 8

2 8 792 04 396 0

( 22)( 18) 018 or 22

n nn n n n

n nn nn n

n nn

− =+

⇒ + − = +⇒ = +⇒ + − =⇒ + − =⇒ + − =⇒ = −

So the number of goats was 18. On the web Quadratic equations applied to problem solving are covered in a page from Purplemath at www.purplemath.com/modules/quadprob.htm.


Assessment


Curve A is the reflection in the x-axis of y = x2.

What is the equation of curve A?

A fountain at ground level sprays out jets of water. Each jet is a parabola. The jet that sprays the farthest has equation y = –x2 + 8x – 15. Factorise this expression.

Hence find (a) where the fountain jet is positioned in this xy-coordinate system and (b) how far from the fountain jet the water hits the ground.

Calculate the greatest height that the water reaches.

Assessment Set up activities that allow students to demonstrate what they have learned in this unit. The activities can be provided informally or formally during and at the end of the unit, or for homework. They can be selected from the teaching activities or can be new experiences. Choose tasks and questions from the examples on the right to incorporate in the activities. Huda throws a ball to Mariam who is standing 20 m away.

The ball is thrown and caught at a height of 2.0 m above the ground.

The ball follows the curve with equation y = 6 + c(10 – x)2, where c is a constant. Calculate the value of c by substituting x = 0, y = 2 into the equation.

An n-sided polygon has 1⁄2 n(n – 3) diagonals. A polygon has 104 diagonals. How many sides does it have?

y = (x – 3)2 + 5 is a quadratic function of x. What is the minimum value of this function and for what value of x does it occur? What is the maximum range of the function? Give the equation of the axis of symmetry of the function. Write an alternative form for the equation defining the function. Sketch the graph of this function.

The diagram shows a rectangular field. Yasir walks round the field along two straight paths from A to B and from B to C. He walks a total of 160 metres from A to C. Lara walks in a straight line diagonally across the field, from A to C. She walks a total of 120 metres.

Let the distance AB be x metres. Write down an equation for x and show that it simplifies to

x2 – 160x + 5600 = 0

Solve this equation to find the distance AB, given that it is shorter than the distance BC.

MEI

Unit 11A.3

165 | Qatar mathematics scheme of work | Grade 11 advanced | Unit 11A.4 | Trigonometry 1 © Education Institute 2005

GRADE 11A: Trigonometry 1

Using sine, cosine and area rules

About this unit This is the first of two units on trigonometry for Grade 11 advanced. It builds on the work on trigonometry in Grade 10 and extends problems to the case of triangles which are not right-angled.





Previous learning To meet the expectations of this unit, students should already know and use the standard trigonometric ratios to find the remaining sides of a right-angled triangle given one side and one angle, or to find the angles given two sides.

Expectations By the end of the unit, students will use trigonometry to solve practical and theoretical problems, breaking them down into smaller tasks. They will know and use the sine and cosine rules, and will calculate the area of a triangle using ½ ab sin C.

Students who progress further will solve more complex problems.

Resources The main resources needed for this unit are: • overhead projector (OHP) • Internet access, computer and data projector • graph plotting software such as:


• dynamic geometry system (DGS) such as: Geometer’s Sketchpad (see www.keypress.com/sketchpad) Cabri Geometrie (see www.chartwellyorke.com/cabri.html)

• computers with Internet access, graph plotting and dynamic geometry software for students


Key vocabulary and technical terms Students should understand, use and spell correctly: • sine (sin), cosine (cos), arc sine (arcsin), arc cosine (arccos)

UNIT 11A.4 9 hours






11A.1.4 Break down complex problems into smaller tasks.

10A.6.5 Know the standard trigonometric ratios and their standard abbreviations, e.g. for sine of θ, given an angle θ in a right-angled triangle, and use these ratios to find the remaining sides of a right-angled triangle given one side and one angle or to find the angles given two sides.

11A.8.2 Know and use the sine rule and the cosine rule to solve triangles

11A.8.3 Solve triangle problems in two and three dimensions.

2 hours

Calculating areas of triangles

2 hours

The sine rule

2 hours

The cosine rule

3 hours

Problem solving

11A.8.4 Calculate the area of a triangle using ½ ab sin C.

Unit 11A.4


Activities


2 hours

Calculating areas of triangles

Calculate the area of a triangle using ½ ab sin C.

Symmetries of sine and cosine graphs: the area rule

Practical activity Either with Autograph or with a graphics calculator (in degree mode), get students to produce a display of the graphs of y = sin x and y = cos x over the domain 0° ≤ x ≤ 180°.

Point out the symmetries. Ask students to speculate on what happens if the domain is extended further; they can of course verify any conjecture with a further display.

Demonstration of symmetries in Autograph


Class discussion From the graphs, establish that: sin x° = sin (180 – x)° and cos x° = –cos (180 – x)° Ask students to confirm these results using random values on a calculator (for example, by comparing 40° and 140°). Consider extension to sines and cosines of angles in the range zero to four right angles if appropriate for more able students.

Sketch a case of an acute-angled triangle with two sides and an included angle given. Ask the class how this information can be used to calculate the area of the triangle. Remind students of the conventions for naming angles and sides of triangles (small letters for sides and corresponding capital letters for opposite angles).

Do this by providing a number of sketched triangles with measurements marked and getting students to verbalise the data in the correct format.

Establish the area rule, ½ ab sin C, using the conventional notation.

Show that the relationship sin x° = sin (180 – x)° enables this to work even when angle C is bigger than a right angle, and that the case where angle C is a right angle reduces to the standard formula ½ bh.

The naming convention for sides: corresponding sides and opposite angles use the same letter.

Exercises Give practice on this rule in simple cases.

Examples • Find the area of PQR in which PQ = 4 cm, QR = 5 cm and ∠PQR = 40°. • XYZ has area 49 m2 and two sides XY = 10.7 m and YZ = 13.5 m. Find the two possible

sizes of ∠XYZ, and sketch the two triangles.

On the web The BBC's Bitesize site covers the work of the trigonometrical formulae at www.bbc.co.uk/schools/gcsebitesize/ maths/shapeh/areaofatrianglerev1.shtml.

Unit 11A.4



2 hours

The sine rule Know and use the sine rule and the cosine rule to solve triangles.

Investigation Provide two examples of triangles in which two angles and one side are specified, making essential use of the naming convention. (For example, in ABC, a = 5 cm, ∠B = 40°, ∠C = 55°; in XYZ, ∠X = 20°, ∠Y = 140°, x = 17 cm.)

Get students to: • sketch these, with the given information clearly shown; • investigate what further information can be determined from these starting points.

Students should be able to see that they can calculate: • the third angle in each case; • the area; • at least one of the unknown sides.

Class discussion Draw ABC in which ∠B = 55° and ∠C = 40°, and BC = 5 cm. Do this both as a sketch on the board or OHT, with an indication of how it would be done exactly as a construction, and also in a dynamic geometry system (DGS), where the unknowns can be measured by using the software’s capabilities. Challenge students to emulate the software, i.e. to do it by calculation.

The diagram was produced in DGS by creating line segment BC and choosing the two unnamed points so that the angles have the right magnitudes (essentially as a process of construction would). The software then measures the lengths of the segments AC and AB.

Ask for proposals of how to calculate the remaining sides; if the answer is not forthcoming, ask which altitude may be calculated from the given information. (In the example of ABC above, it is possible to calculate an altitude BH, where H lies on AC.) Discuss this calculation, to see how it can lead to the calculation of b. Ask students: • Is it also possible to calculate c, and to calculate the area?

Avoid giving practice in this as a routine, since it would reinforce what is ultimately an inefficient procedure. Nevertheless, students should discover the way that the sine rule is proved, namely:

sin sin

sin sin

sinfor the purposesin

sin of calculating to establish

sin sin sin the sine rule

hA h c AchC h a Ca

haC

c AC a

a c bA C B

= ⇒ =

= ⇒ =

⇒ =

⎧= ⎨

⎩⎧⎛ ⎞⇒ = = ⎨⎜ ⎟

⎝ ⎠ ⎩



Rework the calculation with a, B, C instead of the numbers given, so that the sine rule emerges

as sin sinb cB C

= .

Invite students to suggest what would result if the same procedure were used to calculate a (see the notes on the right).

Demonstrate another case to show the routine of using the sine rule in such a situation.

Establish the procedures: • write down the sine rule; • tick the letters which represent known measures (to establish what may be found); • do the calculations.

More able students should be encouraged to derive sinsinb Aa

B= before substituting.

The sine rule can also be established using the circle geometry results of Unit 10A.2. Consider

ABC and its circumcircle centre O.

An isosceles triangle is formed by the two radii OA and OB (= r) and the side AB = c of ABC. Hence the two angles marked x are each equal to ∠C (∠s on same arc). Then, in AOH,

1⁄2 c = HB = r sin C, i.e. 2sinc rC

=

This argument can be repeated for each pair (b, B) and (a, A), so the sine rule follows.

Exercises Get students to practise this technique on several examples. Use the opportunity to reinforce clear presentation: • draw a diagram first with letters and information marked; • set out calculations clearly; • consider appropriate accuracy for the final answers.

Class discussion Demonstrate the use of the sine rule to find an unknown angle (for example, in ABC, a = 5 cm, b = 7 cm, ∠B = 50°; find ∠A), taking care to avoid the ambiguous case (by specifying cases where the angle to be determined is not opposite the largest side).

Exercises Set questions which use this technique on several similar cases.

More able students may show interest in the ambiguous case and how the calculation ties in with the process of construction, i.e. how two different triangles can be realised from the same data in each case.



2 hours

The cosine rule Know and use the sine rule and the cosine rule to solve triangles.

Working up to and using the cosine rule

Investigation

Ask students to sketch triangles in each of these cases: • two sides and an included angle are given;

(for example in ABC, b = 7 cm, c = 6 cm, ∠A = 65°); • three sides are given

(for example in XYZ, x = 6 cm, y = 5 cm, z = 4 cm).

Ask students to consider what further information can be readily calculated (the area in the first case and nothing in the second).

Class discussion Take the first case of triangle ABC where two sides and an included angle are specified. Ask: • What construction was helpful when considering the sine rule?

If no answer is forthcoming, introduce one of the two altitudes (BH or CK). Derive the cosine rule by applying Pythagoras’ theorem to the two resultant right-angled triangles.

Spend some moments considering the structure (symmetry and arrangements of letters) of this formula.

More able students should be able to write down equivalent forms for b2 and c2, although less confident students at this stage may prefer to have all such equivalent forms explicitly available to them to make a choice.

Round this discussion off by showing that the cosine rule reduces to Pythagoras’ theorem in the case of an angle that is 90°.

Note that care is needed here to avoid straying into difficulties. The method to use is:

2 2 2

2 2 2

2 2 2

2 2 2

2 2 2

2 2

( )22

22 cos

b x hc a x h

a ax x hc b a ax

c a b axa b ab C

= += − += − + +

⇒ − = −⇒ = + −

= + −

Stress that at the next-to-last line the whole right-hand side is known. It may not be obvious to students that x too is known; give them the chance to volunteer how to calculate it by elementary trigonometry.

Exercises Set questions which require use of the cosine rule to calculate unknown sides. More able students should be able to manage √(b2 + c2 – 2bc cos A) with no intermediate steps.

In later questions, extend the work into: • two-stage calculations, such as those that require calculation of the area; • word problems.

Although this appears to be an exercise similar to that on the sine rule, students will have difficulty at first in obtaining the correct answer from a calculator; the common mistake (when calculating a2) is to enter b2 + c2 – 2bc and erroneously carry out a calculation; check early that that this pitfall has been avoided.



Class discussion Starting from a presentation of the sine and cosine rules, ask students to consider the case of a triangle with three sides given. Ask: • Which formula can be used to yield results?

This is an opportunity to help students to appreciate which formula is of value in any given context. Less confident students will only make progress if they realise that the sine rule is of no value (in a particular case) before trying the cosine rule. Demonstrate: • how the cosine rule can be rearranged (see the notes on the right); • the three equivalent forms; • how to use the appropriate version to find an unknown angle when three sides are given.

Students should be familiar with the formula

2 2 2

cos2

b c aAbc

+ −=

and its two equivalents.

Exercises Set exercises in two stages: • cases of three sides given; • miscellaneous examples where the solution depends only on a single use of the sine or

cosine rule.

3 hours

Problem solving Break down complex problems into smaller tasks.

Solve triangle problems in two and three dimensions

Using the sine and cosine rule to solve problems

Class discussion Review the three formulae (area rule, sine rule, cosine rule) and the naming conventions. Present a problem that involves all three as an illustration of strategy. Indicate to students the importance of carrying forward an accurate (rather than truncated) answer in successive parts of the question to make the final answer(s) sufficiently accurate.

Example In ABC, a = 4 cm, b = 6 cm, c = 7 cm; find one of the angles, and hence find the area.

As a general rule, where it is not clear how to proceed, write down the sine rule to see if it can help. If not, try the cosine rule.

Solution The sine rule cannot be used here.

2 2 2

2 2 2

arccos2

6 7 4arccos2 6 7

34.771...34.8 to 1 d.p.

b c aAbc

+ −=

+ −=× ×

==

The interim value should be stored in one of the calculator’s memories. For the area we have: 12 sin 0.5 6 7 sin34.771...

11.97...12.0 to 3 s.f.

bc A = × × ×==

So the area is 12.0 cm2.

On the web Mathcentre has worked examples of the rules at www.mathcentre.ac.uk/students.php/ all_subjects/trigonometry/sine_cosine/ resources/52.



Exercises Set miscellaneous problems of increasing complexity on the area, sine and cosine rules. Include: • miscellaneous one-stage calculations where the main emphasis is choice and use of the

correct rule; • two-stage calculations where the first step is obvious or indicated; • problems where strategy is important; • problems posed in words to develop problem solving skills.

Less confident students should be able to progress beyond one-stage problems here.

Problems in three dimensions

Class discussion Discuss a problem in three dimensions.

Example A solid with four sides – a tetrahedron – has vertices A, B, C and D. AB = 4 cm, AC = 5 cm, AD = 6 cm, BD = 7 cm, ∠ABC = 80°, ∠DAC = 40°. Sketch the figure and mark the distances and angles given. Find: • ∠ACB; • ∠ABD; • BC; • DC; • ∠DBC; • the total surface area of the figure.

Stress the main points of good technique: • a good diagram; • the repeated drawing of plane figures which represent each triangle worked on; • taking a strategic view of the problem.

Steps of the solution • Draw ABC and use the sine rule to calculate ∠ACB – it must be acute. • Draw ABD and use the cosine rule to calculate ∠ABD. • Use ABC again to calculate ∠BAC. • Use ABC again and the cosine rule to calculate BC. • Draw ADC and use the cosine rule to calculate DC. • Use the cosine rule to calculate ∠DBC. • Calculate the area of each triangular side and add those values together.



Exercises Set problems with three-dimensional situations. Start with one-stage calculations to begin with, moving gradually to full-scale problem solving as in the worked example above or the example below.

Example:

The Great Pyramid of Cheops in Egypt is built on a square base with side 230 metres. Each face of the pyramid is at 52° to the horizontal. Calculate the height of the pyramid. Calculate the inclination of an edge of the pyramid to the horizontal.

The Great Pyramid of Cheops at Giza

(source: www.kingtutshop.com)


Assessment


Show that Pythagoras’ theorem is a special case of the cosine rule.

A triangle has its three angles in the ratio 2 : 3 : 4. Find to two significant figures the ratio of the lengths of its sides.

A helicopter at airfield A received a distress call from a boat. The position of the boat, B, was given as 147 km from the airfield, on a bearing of 072°. A man on the boat is flown to hospital. Calculate the distance the helicopter travelled from the boat to the hospital at H.


On another occasion the helicopter travelled from the airfield on a bearing of 218° to fly a pregnant woman at W to the hospital. The helicopter then flew on a bearing of 081° to the hospital, H. Calculate the distance the helicopter travelled from where it picked up the woman to the hospital.

The two sides of a canal are straight, parallel and the same height above the water level. Jana and Sharifa want to find the width of the canal. They measure 100 m on the canal bank and stand facing each other at points J and S. Jana measures the angle she turns through to look at a post, P, as 25°. Sharifa measures the angle she turns through to look at the post as 15°. Calculate the width of the canal.

The diagram shows two concentric circles. OP = 3.0 cm and OQ = 5.0 cm.

a. Calculate the shaded area.

b. Calculate the length of PQ when the angle POQ is 130°. Give your answer to a suitable degree of accuracy.

R and S lie on the same concentric circles as P and Q.

c. The area of triangle ROS is 5 cm2. Find the possible sizes of angle ROS.

MEI

Unit 11A.4

175 | Qatar mathematics scheme of work | Grade 11 advanced | Unit 11A.5 | Probability 1 © Education Institute 2005

GRADE 11A: Probability 1

Independent and dependent events

About this unit This unit is the first of two on probability for Grade 11 advanced. In it students meet elementary ideas of probability and how to formalise them into mathematical notation, building on the work on elementary probability in Grades 8 and 9.





Previous learning To meet the expectations of this unit, students should already be able to use mathematics to model and predict the outcomes of real-world applications. They should be able to select statistics and a range of charts, graphs and tables to present findings.

Expectations By the end of the unit, students will use a simple mathematical model to calculate, for a particular set of events, the theoretical probability of obtaining a particular outcome for a random variable associated with those events. They will calculate probabilities of single and combined events. They will use tree diagrams to represent and calculate the probabilities of compound events when the events are independent and when one event is conditional on another.

Students who progress further will readily relate the definitions of independence and exclusive events to practical situations in order to apply their knowledge.

Resources The main resources needed for this unit are: • overhead projector (OHP) • Internet access, computer and data projector • computers with Internet access for students • scientific calculators for students • spinners, dice, coins, drawing pins for simple probability demonstrations,

or simulations of these

Key vocabulary and technical terms Students should understand, use and spell correctly: • event, exclusive, independent, exhaustive • experiment, random variable, outcome • empirical probability, theoretical probability, conditional probability, relative

frequency, expected frequency, symmetry

UNIT 11A.5 10 hours



10 hours SUPPORTING STANDARDS Grade 8 and 9 standards



9.8.5 Use relative frequency as an estimate of probability and use this to compare outcomes of experiments.

11A.1.2 Use mathematics to model and predict the outcomes of real-world applications; compare and contrast two or more given models of a particular situation.

12AS.1.2 Use mathematics to model and predict the outcomes of substantial real-world applications, and to compare and contrast two or more given models of a particular situation.

12AQ.1.2 Use statistical techniques to model and predict the outcomes of statistical situations, including real-world applications; work to definitions and perform appropriate tests.

11A.1.10 Approach a problem systematically, recognising when it is important to enumerate all outcomes.

12AQ.1.10 Approach complex problems systematically, recognising when and how it is important to enumerate all outcomes.

11A.12.1 Know that all probability values lie between 0 and 1, and that the extreme values correspond respectively to impossibility and certainty of occurrence.

12AQ.11.1 Know that all probability values lie between 0 and 1, and that the extreme values correspond respectively to impossibility and certainty of occurrence; calculate probabilities.

12AQ.11.2 Know that all possible outcomes for an experiment form the sample space for that experiment; use the sample space to calculate probabilities for each outcome.

11A.12.2 Understand that a random variable has a range of values that cannot be predicted with certainty, and investigate common examples of random variables; measure the empirical probability (relative frequency) of obtaining a particular value of a random variable.

12AQ.11.3 Understand that a random variable has a range of values that cannot be predicted with certainty; investigate common examples of random variables; measure the empirical probability (relative frequency) of obtaining a particular value of a random variable.

8.8.9 Use problem conditions to calculate theoretical probabilities for possible outcomes.

3 hours

Empirical probability

2 hours

Mutually exclusive and exhaustive events

2 hours

Independent events

3 hours

Probability trees

9.8.7 Compare experimental and theoretical probability in different contexts.

11A.12.3 Use a simple mathematical model to calculate, for a particular set of events, the theoretical probability of obtaining a particular outcome for a random variable associated with those events.

12AQ.11.4 Know that a probability distribution for a random variable assigns the probabilities of all the possible values of the variable and that these values total to 1; use a simple mathematical probability distribution to calculate, for a particular set of events, the theoretical probability of obtaining a particular outcome for a random variable associated with those events.

11A.12.5 Understand when two events are mutually exclusive, and when a set of events is exhaustive; know that the sum of probabilities for all outcomes of a set of mutually exclusive and exhaustive events is 1, and use this in probability calculations.

12AQ.11.8 Understand when two events are mutually exclusive, and when a set of events is exhaustive; know that the sum of probabilities for all outcomes of a set of mutually exclusive and exhaustive events is 1, and use this in probability calculations.

Unit 11A.5


10 hours SUPPORTING STANDARDS Grade 8 and 9 standards



11A.12.6 Know that when two events A and B are mutually exclusive the probability of A or B, denoted by P(A ∪ B), is P(A) + P(B), where P(A) is the probability of event A alone and P(B) is the probability of event B alone.

9.8.6 Know that if A and B are mutually exclusive, the probability of A or B is the sum of the probabilities of A and of B.

11A.12.7 Know that two events A and B are independent if the probability of A and B occurring together, denoted by P(A ∩ B), is the product P(A) × P(B).

12AQ.11.9 Know that: • when two events A and B are mutually exclusive

the probability of A or B, denoted by P(A ∪ B), is P(A) + P(B), where P(A) is the probability of event A alone and P(B) is the probability of event B alone;

• two events A and B are independent if the probability of A and B occurring together, denoted by P(A ∩ B), is the product P(A) × P(B);

• when two events A and B are not mutually exclusive the probability of A or B, denoted by P(A ∪ B), is P(A ∪ B) = P(A) + P(B) – P(A ∩ B), where P(A) is the probability of event A alone, P(B) is the probability of event B alone and P(A ∩ B) is the probability of both A and B occurring together.

9.8.10 List systematically all the possible outcomes of an experiment.

11A.12.8 Use tree diagrams to represent and calculate the probabilities of compound events when the events are independent and when one event is conditional on another.

12AQ.11.10 Use tree diagrams to represent and calculate the probabilities of compound events when the events are independent and when one event is conditional on another.

11A.12.9 Know that in general if event B is dependent on event A, then the probability of A and B both occurring is P(A ∩ B) = P(A) × P(B|Α), where P(B | Α) is the conditional probability of B given that A has occurred.


Activities


Probability and long-term relative frequency Ask students what they mean by stating that the probability of obtaining a head when tossing a coin is 1⁄2. From the discussion, draw out: • the common misunderstanding that if one toss is a head the next is more likely to be a tail; • the correct perception that any experimental justification requires a large number of tosses; • the idea that the symmetry of a coin justifies the claim.

To establish that a large number of tosses is required to be able to state with confidence that probability is equal to relative frequency may require an experiment to justify it. Simulations are a way to generate a lot of data, e.g. Coin tossing from nlvm.usu.edu/en/nav/vlibrary.html, or Adjustable spinner from illuminations.nctm.org/tools/index.aspx

Then establish that there are two ways of modelling such a long-term relative frequency: it is necessary to do an experiment or to use symmetry.

An example of the first situation is the experiment of throwing drawing pins into the air and counting the proportion which land pin down. It is not possible to predict the outcome by considerations of symmetry; indeed, different types of drawing pin may lead to different results. However, the symmetry argument can be used with coins, playing cards, spinners, etc.

Finish the discussion (or round off the experimental session) by defining expected frequency as probability multiplied by number of experiments.

To fix ideas of probability if required, do an experiment with coins or drawing pins. Ask students to record in groups the proportion of coin tosses that produce heads.

If students work in groups make sure that they use identical materials. Then bring together the results with a cumulative table like the one below. .

No. of tosses 20 40 60 80 100

Proportion of heads

When plotted on a graph these values should (usually!) show that the value of the proportion settles on 0.5 approximately as the number of tosses progressively increases. This is an impression that students must acquire if they are to grasp the interpretation of probability. The resulting graph can easily be displayed on a graphics calculator.

The same approach can be taken if students are tossing drawing pins, with results showing how many land pin down recorded in a table.

3 hours

Empirical probability Use mathematics to model and predict the outcomes of real-world applications; compare and contrast two or more given models of a particular situation.

Approach a problem systematically, recognising when it is important to enumerate all outcomes.

Know that all probability values lie between 0 and 1, and that the extreme values correspond respectively to impossibility and certainty of occurrence.

Understand that a random variable has a range of values that cannot be predicted with certainty, and investigate common examples of random variables; measure the empirical probability (relative frequency) of obtaining a particular value of a random variable.

Use a simple mathematical model to calculate, for a particular set of events, the theoretical probability of obtaining a particular outcome for a random variable associated with those events.

Exercises Set exercises which bring out three issues: • calculation of probabilities by symmetry (such as for coins, playing cards, spinners); • interpretation of probability (such as comparisons with real outcomes like small and large

numbers of coin tosses); • prediction of expected frequencies.

In doing this see that students: • do not express probabilities as percentages; • use exact arithmetic when appropriate; • focus on the fact that probability necessarily lies between 0 and 1 and that the two extreme

values have interpretations in terms of impossibility and certainty.

On the web Mathgoodies has a page on elementary probability theory at www.mathgoodies.com/lessons/vol6/ intro_probability.html.

Unit 11A.5



2 hours

Mutually exclusive and exhaustive events Understand when two events are mutually exclusive, and when a set of events is exhaustive; know that the sum of probabilities for all outcomes of a set of mutually exclusive and exhaustive events is 1, and use this in probability calculations.

Know that when two events A and B are mutually exclusive the probability of A or B, denoted by P(A ∪ B), is P(A) + P(B), where P(A) is the probability of event A alone and P(B) is the probability of event B alone.

From intuition to precise language

Class discussion Introduce students to the ideas of exclusive and exhaustive events. These are ideas of which they probably already have an intuitive grasp, and the main task is to establish and use the vocabulary. Do this by using many examples; test students’ ability to recognise the right word to use in many different contexts.

Examples • Toss a coin and consider the events: – the coin is heads – the coin is tails (exclusive and exhaustive). • Shuffle a pack of playing cards, draw a card at random, and consider the events: – the card is black – the card is a face card (neither exclusive nor exhaustive). • Throw a dice and consider the events: – the number shown is even – the number shown is 5 (exclusive but not exhaustive). • Spin a spinner with the digits 1, 2 and 3, and consider the events: – the score is prime – the score is odd (exhaustive but not exclusive).

Exercises Set questions that pose similar situations to those listed above, and require students to classify the events in the same way.

Additive probability law

Class discussion Round this part of the unit off by establishing the additive probability law: P(A ∪ B) = P(A) + P(B) when A and B are mutually exclusive

Associate this idea with a suitable probability space diagram (you may want to remind them of their work on sets and Venn diagrams in Grade 10).

A good example to illustrate the cases where the rule does and does not work is that of rolling two dice (according to whether the events selected are or are not mutually exclusive).



Represent the probability space by a 6 by 6 grid, and consider the pair of events: • the first dice is 4 • the second dice is 6

These two do not overlap, so P(4 or 6) = P(4) + P(6). By contrast, P(4 on first or 6 on the second) ≠ P(4 on the first) + P(6 on the second).

More able students may appreciate the fuller version of the additive law, easily justified – at least visually – by a Venn diagram of the probability space. More able students also will see the analogy between an event and a subset implicit in this.

The full version of the law which holds in all cases is P(A ∪ B) = P(A) + P(B) – P(A ∩ B).

Exercises Give more practice based on the same ideas but with the addition of probability calculations.

Example Several balls are placed in a bag. The balls are identical except for bearing the numbers 2 to 8; every ball has one number and no two have the same. One ball is drawn at random. • How many balls are there in the bag to start with? • What is the probability that the drawn ball has a 2 on it? • What is the probability that the number on the ball drawn is prime? • What is the probability that the number on the ball drawn is odd? • What is the connection between the three previous answers?

On the web Combining probabilities is a topic on Waldo’s Interactive Maths pages at www.waldomaths.com; follow the prompt from 14–16.

2 hours

Independent events Know that two events A and B are independent if the probability of A and B occurring together, denoted by P(A ∩ B), is the product P(A) × P(B).

Precision without confusion

Class discussion Introduce the idea of independent events. Do this by focusing on successive events where the probability of the second is (or is not) affected by the first.

Examples • Removing balls from a bag with replacement and shaking each time leads to independent

events. • The same without replacement leads to dependent events.

Do these experiments with some actual balls in a bag (e.g. 9 red, 3 blue). Follow with more thought-provoking cases such as drawing a card from a pack and considering the events:

Independent Dependent

the card is black the card is black

the card is a face card the card is a spade

Then establish the multiplication law: P(A ∩ B) = P(A) × P(B)

provided that the events A and B are independent.

Illustrate this with the case of the drawn playing card just considered.

Students may remark that exclusive events can be represented in a Venn diagram. The same is not true for independence, but the idea anticipates probability tree diagrams.

The playing card example of independence shows clearly how this law works, despite a feeling on the part of students that since the first event in that case does affect the possible outcomes for the second, the events may in fact not be independent.



Exercises Set questions which explore these ideas in simple terms. Include: • simple probability calculations which exploit the idea of independence, such as: – finding the probability of drawing a spade from a pack of cards and rolling a 6 on a dice; – finding the probability of obtaining ten heads in ten tosses of a coin; – finding the probability of guessing correctly a four-digit number (consisting of four digits,

including zero, in any order); • testing for independence, for example: – when rolling a dice do you think the two events (getting a prime number, getting a multiple

of 3) are independent? Why?

3 hours

Probability trees Use tree diagrams to represent and calculate the probabilities of compound events when the events are independent and when one event is conditional on another.

Know that in general if event B is dependent on event A, then the probability of A and B both occurring is P(A ∩ B) = P(A) × P(B | Α), where P(B | Α) is the conditional probability of B given that A has occurred.

Problem solving

Class discussion Set students a couple of questions from a standard exercise on probability trees, but without further assistance.

Example Feruk has ten coins in his pocket, six identical bronze ones and four identical silver ones. He draws two coins from his pocket. What is the probability that one is bronze and the other silver?

Able students may manage an answer to this but some may not. Of those who do, many will get 4⁄15 rather than the correct 8⁄15. This may well provoke argument. This motivates the introduction of a tree diagram to clarify the situation.

Introduce the tree diagram for the question, stressing the correct convention for labelling the branches with probability values and events. Demonstrate the multiplication rule for finding the probability associated with any particular path, and show the two paths associated with the question being considered.

On the web GCSE Bitesize has a section on conditional probabilities at www.bbc.co.uk/schools/ gcsebitesize/maths/datahandlingh/ probabilityhrev1.shtml.

Exercises Set examples that become progressively harder. Include: • situations in which two successive independent events occur (such as drawings with

replacement, successive sets of traffic lights, etc.); • situations in which two successive dependent events occur; • more complicated three-stage problems.



Class discussion Round off this section by introducing the conditional probability rule: P(A ∩ B) = P(A) × P(B | Α)

Point out that students have now grown familiar with it, as they used it unknowingly in trees. The only new point to make is the notation: P(B | Α) is the probability of B given that A has already occurred.

This diagram prefigures the binomial theorem in probability when it is extended to n stages. Make a connection with the work of Unit 11A.1 if that seems appropriate.

Problem solving In a television quiz show, a contestant has to choose one of three doors. Two conceal a booby prize of no value, while the other conceals a large sum of money that the contestant hopes to win. Once the contestant has made a choice of door, but before that door is opened, the host opens one of the other two doors to reveal a booby prize, and then offers the contestant the chance to change their mind, that is to choose the remaining door instead of the first choice. Which should the contestant do?

If time permits give the question to students to discuss. If you need to give a hint, ask students to work out the probability of winning on each of the two strategies separately, i.e. • to stick to the original choice; • to change. Whichever gives the greater probability is the correct answer.

Using ICT Try some of the interactive probability problems on www.cut-the-knot.org: for example, the Lewis Carroll problem.

A bag contains a counter, known to be either white or black. A white counter is put in, the bag is shaken, and a counter is drawn out, which proves to be white. What is now the chance of drawing a white counter?


Assessment


Assume that it is equally likely for a woman to give birth to a girl as it is for her to give birth to a boy. What is the probability that a woman with six children has four girls and two boys? What is the probability that if another woman has four children they are all boys? A woman with three daughters is going to have a fourth child. What is the probability that the fourth child will be a boy?

The probability of dying of cancer is 1⁄3. What is the probability that if three people are chosen at random two of them will die of cancer? What is the probability that none of them will die of cancer?

Mona has a chance of 1 in 4 of passing on a particular genetic condition to one of her three children. Calculate the probability that two of the children will inherit the condition. What is the probability that none of her children will inherit the condition? What is the probability that at least one of her children will inherit the condition?


A computer game has nine circles arranged in a square. The computer chooses circles at random and shades them black. At the start of the game, two circles are to be shaded black.

a. Show that the probability that both circles J and K will be shaded black is 1⁄36.

b. Halfway through the game, three circles are to be shaded black. Here is one example of the three circles shaded black in a straight line. Show that the probability that the three circles shaded black will be in a straight line is 8⁄84.

c. At the end of the game, four circles are to be shaded black. Here is one example of the four circles shaded black forming a square. What is the probability that the four circles shaded black form a square?

.

In a class of 35 students, the probability that a student picked at random is taller than 1.8 metres is 0.2 and the probability that the student wears spectacles is 0.3.

What is the probability when three students are chosen at random that two are over 1.8 metres in height and that one of them wears spectacles?

On a road there are two sets of traffic lights. The traffic lights work independently. For each set of traffic lights, the probability that a driver will have to stop is 0.7.

a. A woman is going to drive along the road. What is the probability that she will have to stop at both sets of traffic lights? What is the probability that she will have to stop at only one of the two sets of traffic lights?

b. In one year, a man drives 200 times along the road. Calculate an estimate of the number of times he drives through both sets of traffic lights without stopping.

Unit 11A.5



Adel makes two clay pots. Each pot is fired independently. The probability that a pot cracks while being fired is 0.03.

a. Calculate the probability that both of Adel’s pots crack while being fired.

b. Calculate the probability that only one of Adel’s pots cracks while being fired.

c. Adel has enough clay for 80 pots. He receives an order for 75 pots. Does he have enough clay to make 75 pots without cracks? Explain your answer.

On a tropical island the probability of it raining on the first day of the rainy season is 2⁄3.

If it does not rain on the first day, the probability of it raining on the second day is 7⁄10.

If it rains on the first day, the probability of it raining more than 10 mm on the first day is 1⁄5.

If it rains on the second day but not on the first day, the probability of it raining more than 10 mm is 1⁄4.

You may find it helpful to fill in the tree diagram before answering the questions below.

a. What is the probability that it rains more than 10 mm on the second day, and does not rain on the first?

b. What is the probability that it has rained by the end of the second day of the rainy season?

c. Why is it not possible to work out the probability of rain on both days from the information given?

20 per cent of the population of a country has a particular disease. A test can be given to help determine whether or not people have the disease. The probability that the test is positive for those that have the disease is 0.7. But there is a 0.1 chance that a person who does not have the disease registers positive on the test.

a. Find the probability that an individual selected at random tests positive, but does not have the disease.

b. Another person is chosen at random. Calculate the probability that the test result for this person is positive.



Quadratic equations and inequations

About this unit This is the second of four units on algebra for Grade 11 advanced. It continues the work of formal manipulation of symbols in parallel with the development of graphical models, building on Unit 11A.3.





Previous learning To meet the expectations of this unit, students should already be able to solve quadratic equations exactly, by factorisation, by completing the square and by using the quadratic formula, and to model situations with simple quadratic functions. They should be able to interpret the solution of a pair of simultaneous linear equations.

Expectations By the end of the unit, students will form and manipulate algebraic expressions and formulae, rearrange harder formulae and generate further formulae. They will find approximate solutions of quadratic equations, and of a pair of simultaneous equations, one linear and one quadratic. Through their study of functions and their graphs, and the solution of associated equations, students will appreciate a range of numerical and algebraic applications in the real world. They will continue to use ICT to analyse problems.

Students who progress further will recognise with increasing confidence the graphical significance of algebraic results.



• computers with Internet access and graph plotting software for students • graphics calculators for students

Key vocabulary and technical terms Students should understand, use and spell correctly: • simultaneous, linear, quadratic, root, distinct, coincident, inequation,

inequality • discriminant, strictly non-negative, positive definite

UNIT 11A.6 12 hours








12AS.1.6



11A.2.1 Make appropriate use of knowledge of number sets.

10A.4.11 Solve quadratic equations exactly, by factorisation, by completing the square and by using the quadratic formula.

11A.5.4 Given a quadratic equation of the form ax2 + bx + c = 0, know that: • the discriminant Δ = b2 – 4ac must be non-negative for

the exact solution set in to exist; • there are two distinct roots if Δ is positive and one

repeated root if Δ is zero.

12AQ.6.1 Given a quadratic equation of the form ax2 + bx + c = 0, know that if the discriminant Δ = b2 – 4ac is negative, there are two complex roots, which are conjugate to each other.

11A.5.6 Solve equations and inequalities using algebra or a combination of algebra and graphical representation.

10A.5.19 Recognise a second-order polynomial in one variable, y = ax2 + bx + c, as a quadratic function; plot graphs of such functions.

11A.5.7 Use the graph of the function f(x) = ax2 + bx + c to determine regions where ax2 + bx + c is greater than or less than zero.

2 hours

Quadratic inequations

2 hours

Using graphs to solve harder inequations

4 hours

Solving simultaneous equations by algebraic methods

2 hours

Problem solving in geometry 1

2 hours

Problem-solving in geometry 2

10A.5.14 Interpret the solution set of the simultaneous equations E1 and E2, where E1 and E2 are the equations of two straight lines.

11A.5.8 Find exactly by analytical methods and approximately by graphical methods, the solution set of two simultaneous equations L1 and Q1, where L1 represents a linear relation for y in terms of x, and Q1 a quadratic function of y in terms of x.

12AQ.6.4 Find approximate solutions for the intersection of any two functions from the intersection points of their graphs, and interpret this as the solution set of pairs of simultaneous equations.

10A.5.20 Model situations with quadratic functions of the form y = ax2 + c.

11A.5.9 Solve physical problems modelled simultaneously by two such functions.

11A.8.5 Find the points of intersection of a straight line with a circle by using algebraic substitution from the equation of the straight line into the equation of the circle.

Unit 11A.6


Activities


2 hours

Quadratic inequations Make appropriate use of knowledge of number sets.

Solve equations and inequalities using algebra or a combination of algebra and graphical representation.

Use the graph of the function f(x) = ax2 + bx + c to determine regions where ax2 + bx + c is greater than or less than zero.

Using graphs for inequations

Class discussion Ask students to sketch a quadratic function, say y = x2 – 5x – 6. They should be able to remember the key points about y = ax2 + bx + c from Unit 11A.3: • the shape will be a parabola; • the parabola will be ‘happy’ if a > 0 and ‘sad’ if a < 0; • it will intersect the y-axis at (0, c);

• the axis of symmetry is bxa

= − ;

• its minimum (or maximum) value can be obtained by completing the square (or remembered

as 2

4bca

− , so its turning point is 2

,2 4b bca a

⎛ ⎞− −⎜ ⎟⎝ ⎠

);

• its intersection points with the x-axis are (x1, 0), (x2, 0), where x1 and x2 are the roots of the equation 0 = ax2 + bx + c.

This seems complicated (and is included in full only for completeness) but the basic idea, illustrated in the example below, is trivial.

‘Happy’ and ‘sad’ parabolas illustrated


Consider the equation: y = x2 – 5x – 6

= (x – 6)(x + 1)

The parabola is ‘happy’ because the coefficient of x2 is positive. In the graph on the right, the points where the curve cuts the axes are (–1, 0), (6, 0), (0, –6). (The turning point is (5⁄2, –49⁄4), but that is hardly necessary to sketch the curve.)

Now ask students to solve x2 – 5x – 6 < 0.

To solve the problem requires some thought in interpreting the graph. Point out that the sketch combined with a little algebra effectively solves the equation. The solution is just –1 < x < 6.

Alternatively these questions can bring in set notation. In that case the solution set is: {x: –1 < x < 6, x ∈ }

Adapt the method for a more complicated case (e.g. solve 3x2 – 5x + 2 > 0).

Exercises Set questions of a type similar to those just considered in order to refine students’ factorisation and curve-sketching skills.

Extend the scope by introducing examples which reverse the idea, such as: • Prove that x2 – 4x + 5 > 0 for all values of x.

Solution x2 – 4x + 5 = (x – 2)2 + 1

The curve is ‘happy’ and has a minimum at (2, 1). Since it is above the axis, the inequality holds. Alternatively, the expression is a sum of one non-negative and one positive quantity, so the result is strictly positive.

Unit 11A.6



More of the same There are no new ideas in this section, which is an extension of the work just done.

Introduce it by a worked example.

The graph (in the notes column) shows the curve y = 5x2 – 12x + 3 and the straight line y = 2x – 3. Using the graph: • solve 0 = 5x2 – 12x + 3; • solve 2 = 5x2 – 12x + 3; • solve 0 = 5x2 – 12x – 1; • solve 2x – 3 = 5x2 – 12x + 3; • solve 0 < 5x2 – 12x + 3; • solve 0 < 5x2 – 12x + 5; • find a positive value of x to satisfy 2x – 3 < 5x2 – 12x + 3.

None of the ideas is particularly difficult, although less confident students will find some of it less than obvious. The example brings together ideas of intersecting curves and lines, regions of inequality and parabolas, all of which have been well covered in this unit and previous ones.

2 hours

Using graphs to solve harder inequations Use a range of strategies to solve problems, including working the problem backwards and then redirecting the logic forwards; set up and solve relevant equations and perform appropriate calculations and manipulations; change the viewpoint or mathematical representation, and introduce numerical, algebraic, graphical, geometrical or statistical reasoning as necessary.

Solve equations and inequalities using algebra or a combination of algebra and graphical representation.

Find exactly by analytical methods and approximately by graphical methods, the solution set of two simultaneous equations L1 and Q1, where L1 represents a linear relation for y in terms of x, and Q1 a quadratic function of y in terms of x.

Exercises Set questions which use the graphics calculator’s graphing capabilities (or Autograph’s) to explore these ideas. Points to cover are: • solving quadratic equations and inequations where solutions are approximate and read from

a graph; • solving quadratic and linear equations simultaneously by relating them to a given graph; • relating one quadratic form to another by simple manipulation before using the graph; • turning the problem round by asking which line should be introduced to the graph of a

parabola; • using a graphics calculator to solve a problem (of this type) specified exclusively in algebraic

terms.

These elements give ample scope for bringing a lot of the algebraic skills covered into play in one and the same problem. Nevertheless, maintain the focus strictly on relating algebra to graphs.

Example Use a graphical method (on a graphics calculator) to solve 7 + 4x – 5x2 = 0. By adding one further straight line graph to the display, find the solutions of: • the equation 8 + 3x – 5x2 = 0; • the inequation 8 + 3x – 5x2 < 0.

On the web There is a revision resource on elementary quadratic graphs at www.tech.plym.ac.uk/maths/resources/PDFLaTeX/quad_graphs.pdf.



4 hours

Solving simultaneous equations by algebraic methods Work to expected degrees of accuracy, and know when an exact solution is appropriate.

Find exactly by analytical methods and approximately by graphical methods, the solution set of two simultaneous equations L1 and Q1, where L1 represents a linear relation for y in terms of x, and Q1 a quadratic function of y in terms of x. Solve physical problems modelled simultaneously by two such functions.

Using algebra to solve simultaneous equations

Class discussion Begin by posing a similar problem and requesting an algebraic approach.

Example Solve the simultaneous equations:

2 25 4 4 0

3 2 4x x y

x y− − =

− =

The solution could be drawn from many textbooks. It has several features to emphasise. • Allow students time to suggest an approach. Many will propose elimination of x or y by

adding or subtracting multiples of each equation. Give the opportunity for students to see that this will not work. If, on the other hand, one student suggests – as in the model – using substitution, take the chance to explore the other method since students need to see that it is not fruitful.

• In this case a shortcut would be to substitute for 2y. Again a student may suggest this, and it does make the solution shorter. Demonstrate as in the model, however, that fractions arise more often than not and that handling them takes care and frequent checking.

• The solution is more complex than expected. Give students time to reflect on why that is the case. A sketch of the geometric configuration may help, but beware that this could lead into unknown territory.

• There may be students who suggest substitution for x. In that case take it up and do it on that basis, perhaps after doing it the way shown here. Students will begin to see from both methods that often one is easier and to be preferred!

• Solutions need to be paired so that it is clear that there are just two pairs of coordinates and not four – erroneously including (1, 4), for example.

• In principle, the quadratic equation involved could require the formula (i.e. require approximate roots) or have no solutions. As a rule most textbooks stick to simple quadratics, so students should bear in mind that equations that factorise are what they should normally expect in the absence of a hint.

Model solution

( )

2 2

22

22

2 2

2

2

5 4 4 03 2 4

Solve the linear equation for :2 3 4

3 22

Substitute into the quadratic equation:

35 4 4 2 02

95 4 4 6 4 04

5 4 9 24 16 0

4 20 16 05 4 0

( 4)(

x x yx y

yy x

xy

xx x

xx x x

x x x x

x xx xx x

− − =− =

= −

⇒ = −

⎛ ⎞− − − =⎜ ⎟⎝ ⎠

⎛ ⎞− − − + =⎜ ⎟

⎝ ⎠

− − − + =

− + − =− + =

−

12

1) 01 4

or 4

x xy y

− == =⎫ ⎫

⇒ ⎬ ⎬= − = ⎭⎭

Exercises Set enough questions of the type considered for students to develop fluency and accuracy. Include some applications from practical and scientific contexts.

Contrast these exact methods with the approximate procedures of the previous section.

Round off this section by discussing specific examples which may have led to difficulty (e.g. problems with brackets and fractions).

Get students to use graphics calculators to check their work.

Although the quadratic graphs extend the scope beyond the work covered, the pictures reinforce the novelty of having (usually) two points for solutions.

On the web This topic is covered in Paul’s On-line Math, and taken beyond this level to the next stage. See tutorial.math.lamar.edu/AllBrowsers/1314/ NonlinearSystems.asp.



2 hours

Problem solving in geometry 1 Find the points of intersection of a straight line with a circle by using algebraic substitution from the equation of the straight line into the equation of the circle.

Intersecting lines and circles

Class discussion A special case of the geometrical configurations just considered is that of a line which intersects a circle. Give an example first, and extend the work of the simultaneous equations’ solution to the requirement to sketch the configuration (unaided by a calculator).

For example: • Solve the simultaneous equations:

2 2 4 8 6 0

2 5 0x y x y

x y+ − + − =

− − =

Illustrate your solution with a sketch.

The solution is straightforward, and can be confirmed in (say) Autograph (see notes).

Exercises Set questions on the idea of relating algebra to a geometric configuration as just shown. Extend the scope of these to reinforce ideas from the geometry units.

Example A triangle has two sides which have equations:

2

3 7 26y x

y x=

+ =

Its circumcircle is: 2 2 2 28 0x y y+ + − =

Show that the two lines intersect on the circle.

Find the equation of the third side of the triangle.



2 hours

Problem solving in geometry 2 Develop chains of logical reasoning, using correct mathematical notation and terms.

Given a quadratic equation of the form ax2 + bx + c = 0, know that: • the discriminant

Δ = b2 – 4ac must be non-negative for the exact solution set in to exist;

• there are two distinct roots if Δ is positive and one repeated root if Δ is zero.

Class discussion Consider the condition for tangency. Ask students to decide whether a particular case of a line and circle intersect and what is special about the nature of the intersection; use a case where the line touches the circle (see the example on the right). Students should be able to see that the quadratic that results has two coincident roots and that is the algebraic equivalent of tangency.

Now consider the general problem. • What condition must be satisfied by c if (say) the line is a tangent to the circle in the case

below?

2 2

34 6 4 0

y x cx y x y

= ++ − − − =

The solution is as follows. 2 2

2 2

2 2

2 2

2

2

(3 ) 4 6(3 ) 4 010 (6 22) 6 4 0

If the equation has coincident roots, we must have:(6 22) 40( 6 4) 0

36 40 264 240 484 160 04 24 644 0

6 161 0

3 170

x x c x x cx c x c c

c c cc c c c

c cc c

c

+ + − − + − =⇒ + − + − − =

− − − − =⇒ − − + + + =⇒ + − =⇒ + − =

⇒ = − ±

This can be done experimentally with ICT, so again the correspondence between geometry and algebra is clear. The algebraic method will give an exact answer rather than an approximation.

Bring out clearly the fact that two solutions that result from the condition for tangency correspond to two possible solutions to the geometric problem, i.e. two distinct straight lines.

An example which serves the purpose is:

2 2

2 5 342 28 0

x yx y y

− =+ + − =

Exercises Set questions which explore this idea. The example above involves an unknown c in the line’s equation; use the same idea for determining different cases of tangency, such as: • a line with a variable gradient which passes through a fixed point; • a circle with a variable centre (i.e. just one coordinate unknown); • a circle with a variable constant term (so that the radius is variable).

In all cases encourage exact calculation but with verification using graphics calculators or graph plotting software such as Autograph.


Assessment


The graph shows the curve y = x2 + 4x.

a. Solve the equation x2 + 4x – 2 = 0 using the graph. Give your answers to two decimal places.

b. The equation x2 + 4x + 5 = 0 cannot be solved using the graph. Why not?


Find the solution set of (x – 1)2 ≥ 9.

Solve the inequality | 2x – 3 | ≤ x + 3.

Solve the inequality x2 + x – 6 > 0.

How can you determine how many points of intersection a given straight line y = mx + c has with a given quadratic curve y = ax2 + bx + c?

Find the points where the line 4x – 3y = 0 cuts the circle x2 + y2 = 100.

A circle has two points P (–1, 3) and Q (5, 11) as the ends of a diameter.

a. Find the equation of the line PQ.

b. Find the coordinates of C, the centre of the circle.

c. Find the length of the radius of the circle.

d. Write down the equation of the circle in the form (x – a)2 + (y – b)2 = r2.

e. The line y = 7x – 4 cuts the circle at P and at another point R. Find the coordinates of R.

f. Prove that the line RC is perpendicular to the line PQ.

MEI

Unit 11A.6

193 | Qatar mathematics scheme of work | Grade 11 advanced | Unit 11A.7 | Measures © Education Institute 2005

GRADE 11A: Measures

Rates and compound measures

About this unit This is the only unit on measures for Grade 11 advanced. It consolidates and extends work on mensuration, graphs and rates begun in Grade 10.





Previous learning To meet the expectations of this unit, students should already be able to find perimeters, areas and volumes of common plane shapes and solids, using dimensionally correct units, and to solve straightforward problems involving compound measures.

Expectations By the end of the unit, students will use mathematics to model and predict outcomes of real-world applications, and compare and contrast two or more given models of a particular situation. They will identify and use connections between mathematical topics. They will use geometry and trigonometry to solve practical and theoretical problems. They will solve a range of problems involving compound measures, using appropriate units and dimensions.

Students who progress further will solve more complex problems and will relate the concept of rate to gradient.

Resources The main resources needed for this unit are: • overhead projector (OHP) • Internet access, computer and data projector • computers for students with Internet access • scientific calculators for students

Key vocabulary and technical terms Students should understand, use and spell correctly: • rate, gradient • cone, frustum, curved surface area • displacement, velocity, acceleration

UNIT 11A.7 3 hours



3 hours SUPPORTING STANDARDS

Grade 10A standards CORE STANDARDS Grade 11A standards




11A.1.12 Synthesise, present, discuss, interpret and criticise mathematical information presented in various mathematical forms.

10A.7.1 Find perimeters and areas of rectilinear shapes and volumes of rectilinear solids; find the circumference and area of a circular region, and the surface area and volume of a right prism, cylinder, cone and pyramid, and a sphere, using dimensionally correct units.

11A.11.1 Calculate lengths, areas and volumes of geometrical shapes.

2 hours

Mensuration and problem solving

1 hour

Interpreting graphs

10.A.7.4 Solve problems involving compound measures, including average speed, such as cost per litre, kilometres per litre, litres per kilometre, population density (number of people per unit area), density (mass per unit volume), pressure (force per unit area) and power (energy per unit time).

11A.11.2 Work with compound measures including density, average speed and acceleration, measures of rate (such as rate of growth of income), and population density (number of people per unit area), using appropriate units and dimensions.

11A.5.12 Use physical contexts to plot and interpret graphs of functions.

Unit 11A.7


Activities


2 hours

Mensuration and problem-solving Identify and use interconnections between mathematical topics.

Calculate lengths, areas and volumes of geometrical shapes.

The area and volume of a frustum of a cone

Class discussion Ask students to remind other members of the class about the formulae already studied for areas and volumes. They should remember those studied in Grade 10: • area of the curved surface of a cylinder; • area of the curved surface of a sphere; • volume of a cylinder; • volume of a pyramid; • volume of a cone; • volume of a sphere.

Focus on the cone. Ask students how they would make a paper model of a cone; they should remember that it is done with a sector of a circle. Sketch such a sector and a cone, and ask for the corresponding dimensions. Develop the discussion to show all the interrelations between the various measurements, as shown in the diagrams on the right.

Ask students how to calculate the surface area of the cone, using only dimensions of the cone. If there are no suggestions, pose the simpler question of how to calculate x. As a result of the discussion, derive the expression for the curved surface area of the cone. Ask students how to relate h, s and r; if necessary draw the triangle with h, s and r as sides of a plane figure.

Derivation of the curved surface area of a cone:

2Area where 2

2c πs c πrπsπrs

= × =

=

Continue the discussion with the case of a frustum of a cone. Ask for a method to calculate the volume of a frustum of specified upper and lower radii and height. If necessary, give a hint by drawing the missing part of the cone. Ask students how the dimensions of the cone (radius and height) can be specified; the radius is the base of the frustum, but it may take a moment for students to make the necessary connection with geometrical work. Suggest that similar triangles are the key to the height calculation.

( )

k bh k aka hb kb

k a b hbhbka b

=+

⇒ = +⇒ − =

⇒ =−

Now ask students how to complete the calculation (by subtracting the upper volume from that of the whole cone). As a follow-up question, ask how to calculate the frustum’s curved surface area.

On the web Formulae for the results in this section are quoted on a page of Mathworld at mathworld.wolfram.com/ConicalFrustum.html.


Unit 11A.7



Exercises Set questions which explore the new formulae for a cone and which revise those learned in Grade 10. Extend these to harder problem solving on the same theme.

Examples of harder problems at this stage are: • If a full cylinder of water of height 15 cm and radius 4 cm is poured into an inverted cone of

base radius 7 cm and depth 20 cm, how deep is the frustum of empty space above the water in the cone?

• A bucket in the shape of a frustum of a cone of depth 40 cm, top radius 14 cm and bottom radius 10 cm contains water to a depth of 8 cm. How much does the water rise if a lead sphere of radius 4 cm is immersed in it?

1 hour

Interpreting graphs Use mathematics to model and predict the outcomes of real-world applications; compare and contrast two or more given models of a particular situation.

Synthesise, present, discuss, interpret and criticise mathematical information presented in various mathematical forms.

Work with compound measures including density, average speed and acceleration, measures of rate (such as rate of growth of income), and population density (number of people per unit area), using appropriate units and dimensions.

Use physical contexts to plot and interpret graphs of functions.

Gradients and rates Show students a graph of a train journey, such as the one on the right. Ask them to describe what the various parts of it mean. Ask them to comment on how realistic it may be.

From the discussion draw out the following points: • the various sections correspond to the train’s gathering speed from stationary, maintaining a

constant speed, and then slowing down; • the train’s rate of braking is faster than it accelerates; • the gradient of a section corresponds to acceleration; • positive and negative gradients correspond to speed increasing and decreasing.

Ask students to make a sketch of a better model.

Now make similar points with respect to a (simpler) distance–time graph for completeness.

Remind students of other graphs relating physical quantities, such as mass against volume from experiments, where the gradient of the resultant graph corresponds to a rate measurement, such as density in that case.

Exercises Set questions that: • revise work on rates from earlier grades (such as distance–speed–time calculations, or

interest rates); • relate the concept of rate to graphs; • involve multiple rate calculations.

Examples • Use results from a Hooke’s law experiment to determine the spring constant (as a rate, force

per unit distance of extension). • Calculate the speed at which cable unwinds from a drum rotating at a given rate.

This work makes a connection with the process of determining the gradients of curves in other units, both by drawing tangents and by differentiation. Using ICT Support this work using on-line applets from standards.nctm.org/document/eexamples: 5.2 Understanding distance, speed, and time

relationships using simulation software 6.2 Learning about rate of change in linear

functions using interactive graphs: constant cost per minute

You could also use Savings calculator from nlvm.usu.edu/en/nav/vlibrary.html.


Assessment


On the pinboard, draw a trapezium that has a perimeter of 6 + 4√ 2.

This shape is designed using three semicircles of different sizes.


In the shapes on the right, the radii of the semicircles are 3a, 2a and a.

Find the area of each semicircle, in terms of a and π, and show that the total area of the shape is 6πa2.

Find a when the area is 12 cm2, leaving your answer in terms of π.

A light shade is in the form of a frustum of a right cone. The radius at the top of the shade is 10 cm and the radius at the bottom is 25 cm. Find the surface area of the material used for the light shade.

Jabor sees a flash of lightning. 25 seconds later he hears the sound of thunder.

The speed of light is about 1.1 × 109 km per hour.

The speed of sound is about 1.2 × 103 km per hour.

Calculate how far away Jabor is from the lightning.

A cable car takes passengers to the top of a volcano. It starts from station A and takes 16 minutes to reach station B at the top of the volcano. The average speed of the cable car is 2 metres per second. The cable car is at an angle 25° to the horizontal. Find, to the nearest metre, the height of the volcano as measured from A.

TIMSS Grade 12

Unit 11A.7



Wafa recorded the speed of a car every 10 seconds throughout a short journey from her home to school. She used the data to draw a speed–time graph.

a. Find a point during the journey when the car’s speed was increasing most quickly. Mark this point as P. By drawing appropriate lines on the graph, find the acceleration of the car at point P.

b. Find a point during the journey when the car’s speed was decreasing most quickly. Mark this point as Q. By drawing appropriate lines on the graph, find the acceleration of the car at point Q.

c. The car uses least fuel when it travels at speeds between 20 m/s and 25 m/s. Find an approximate value for the area under the graph for the period when the car was travelling at between 20 m/s and 25 m/s. What does this area represent? Give the correct units.



Miscellaneous functions and their inverses

About this unit This is the third of four units on algebra for Grade 11 advanced, building on Units 11A.3 and 11A.6, Algebra 1 and 2. In the unit, students will advance their use of proportional reasoning and extend their knowledge of functions and graphs to include inverses, symmetries and compositions.





Previous learning To meet the expectations of this unit, students should understand direct proportion. They should be able to generate formulae connecting two or more variables from a physical context, and to rearrange formulae. They should recognise one-to-one and many-to-one mappings as functions and a one-to-many mapping as a relation but not a function. They should know when a graph represents a functional relationship between two variables and when it does not.

Expectations By the end of the unit, students will solve routine and non-routine problems in a range of mathematical and other contexts. They will identify and use connections between mathematical topics. They will recognise when to use ICT and do so efficiently. They will use proportional reasoning to solve problems, including inverse proportion problems. They will form and manipulate algebraic expressions and formulae, and generate further formulae from physical contexts. They will use physical contexts to plot and interpret the graphs of linear, quadratic, cubic, reciprocal, sine and cosine functions, and the modulus function. They will recognise symmetry properties of functions, when they are even or odd, and when they are increasing, decreasing or stationary. They will solve problems using inverse and composite functions.

Students who progress further will develop confidence in relating mathematical topics, in the critical use of ICT and in handling functions in a more abstract way.

Resources The main resources needed for this unit are: • overhead projector (OHP) • Internet access, computer and data projector • spreadsheet software such as Microsoft Excel • graph plotting software such as:


• computers with Internet access, spreadsheet and graph plotting software for students


Key vocabulary and technical terms Students should understand, use and spell correctly: • inverse, proportion, asymptote, undefined • quadratic, cubic, reciprocal • symmetry, odd, even, periodic, point of inflexion • increasing, decreasing, stationary • modulus, sine, cosine • composite, construct, deconstruct

UNIT 11A.8 12 hours








11A.1.14 Recognise when to use ICT and when not to, and use it efficiently. 12AQ.1.16

11A.5.1 Understand the symmetry properties of functions, and when a function is even or odd; sketch and describe the features of polynomial functions up to order 3.

Recognise when to use ICT and when not to, and use it efficiently; use ICT to present findings and conclusions.

11A.5.10 Understand the statement y is inversely proportional to x and set up the corresponding equation y = k / x; know some characteristics, including that x ≠ 0 and that x = 0 is an asymptote to the curve, as is y = 0; study examples of inverse proportionality.

10A.5.6 Translate the statement y is proportional to x into the symbolism y ∝ x and into the equation y = kx, and know that the graph of this equation is a straight line through the origin and that the constant of proportionality, k, is the gradient of this line.

11A.5.11 Use a graphics calculator, including use of the trace function, to show approximate solutions to physical problems requiring the location and physical interpretation of the intersection points of two or more graphs.

12AQ.6.4 Find approximate solutions for the intersection of any two functions from the intersection points of their graphs, and interpret this as the solution set of pairs of simultaneous equations.

2 hours

Cubic graphs

2 hours

The reciprocal function and inverse proportionality

2 hours

Properties of graphs

2 hours

Inverse functions

2 hours

Combinations of functions

2 hours

Using a graphics calculator to solve simultaneous equations

10A.5.1 Investigate a range of mathematical and physical situations to develop the concepts of function, domain and range, recognising one-to-one and many-to-one mappings as functions and a one-to-many mapping as a relation but not a function.

11A.5.12 Use physical contexts to plot and interpret graphs of functions, recognising when the functions are increasing, decreasing or stationary, including: • linear, quadratic and cubic functions; • the reciprocal function y = k / x (x ≠ 0); • the sine and cosine functions; • the modulus function and a range of simple non-standard

functions.

12AQ.7.2 Interpret the numerical value of the derivative at a point on the curve of the function; know that: • when the derivative is positive the

function is increasing at the point; • when the derivative is negative the

function is decreasing at the point; • when the derivative is zero the function

is stationary at the point.

Unit 11A.8





10A.4.13 Generate formulae from a physical context, and rearrange formulae connecting two or more variables.

11A.5.13

10A.5.2 Understand and use the concept of related variables and, in special cases, set up appropriate functional relationships between them.

Find, graph and use the inverse function of those functions given by a one-to-one mapping or restricted to such mappings; know that the graph of the inverse function may be found by reflecting the graph of the function in the line y = x; solve a range of problems using inverse functions.

10A.5.5

Recognise when a graph represents a functional relationship between two variables and when it does not.

11A.5.14 Add, subtract and multiply two functions given in the form y1 = f1(x) and y2 = f2(x); write down, without simplification, the mathematical form for one function divided by another.

12AQ.2.1 Multiply, factorise and simplify expressions and divide a polynomial by a linear or quadratic expression.

11A.5.15 Understand the concept of a composite function and use the notation y = f(g(x)).

12AQ.5.4 Form composite functions and use the notation y = g(f(x)).

11A.5.16 Deconstruct a composite function into its constituent functions, using inverse functions.


Activities


2 hours

Cubic graphs


Sketch and describe the features of polynomial functions up to order 3.

Drawing cubic graphs

Class discussion and practical Graphs of quadratic functions were covered in Unit 11A.6. This section of Unit 11A.8 covers corresponding ground for the cubic function.

Begin by getting students to plot an example of a cubic graph, such as y = x3 – 3x2 – 4x + 2. This has sufficient challenge in the manipulation of signs and has two turning points. If necessary, specify the range and the scale to use.

Use Excel (or another spreadsheet) to display the correct table of values as a check for students, and Autograph to confirm the graph. Check that students are plotting points in the conventional way (dots) and that they are taking trouble to produce a smooth curve.

Ask for the coordinates of both turning points, and for their character as maximum or minimum.

When students have finished, ask for and discuss any other points of interest, for example: • the different directions the graph takes as x → ±∞; • the point of inflexion.

The graph produced by Autograph for

y = x3 – 3x2 – 4x + 2 On the web There is a survey of the possible configurations of a cubic graph at www.maths.abdn.ac.uk/~igc/tch/ ma1002/appl/node31.html.


Investigation

Use graphics calculators to give students experience of different cubic graphs. This exercise can be done just as well in Autograph or other graph plotting software, but calculators are sufficient.

Base this investigation on similar work for the quadratic. The points to draw out in discussion with students are: • the intercept is determined by the constant term; • at large values of x the sign is different for x positive and x negative; • the orientation of the graph depends on the sign of the coefficient of x3; • some curves have no turning points; • all curves cut the x-axis once but some cut it three times; • there is always a point of inflexion; • if the coefficient of x2 is zero the turning point is on the y-axis.

Graphics calculator display of two cubics: y = x3 y = x3 – 4x + 2

Autograph display of the same cubics

Unit 11A.8



Problem solving Determine the equation of the cubic graph in the diagram. The point of inflexion is at (0, 8) and the curve touches the x-axis at (2, 0).

2 hours

The reciprocal function and inverse proportionality

Understand the statement y is inversely proportional to x and set up the corresponding equation y = k / x; know some characteristics, including that x ≠ 0 and that x = 0 is an asymptote to the curve, as is y = 0; study examples of inverse proportionality.

A discontinuous graph

Class discussion and practical

Turn to the graph of 1yx

= . Begin with a discussion of the table of values in order to face

difficulties about x = 0, where y is undefined. Discuss one or two positive values; ask students to contribute the rest. Note the symmetry. Now ask students to plot and draw a smooth curve.

After highlighting the case of x = 0, allow time for students to come to terms with a graph that has a break in it. Explain that this is the visual effect of the lack of definition in this case. Note also the symmetries: anticipate the definition of odd function if appropriate, but certainly note the rotation symmetry of order 2 about the origin.

Extend the work briefly, using Autograph or graphics calculators. Ask students to investigate kyx

= in cases where k is positive and negative.

Round off by showing that in all cases xy = k.

Class discussion Discuss some practical cases of inverse proportion. Examples are: • shares of a sum of money and the number of people who share; • cost of a car trip per passenger and the number of passengers.

In each case discuss a table of values and a graph, to show that these practical cases correspond to the reciprocal graphs just considered. Then go on to the formal method of dealing with these.

A display in Autograph (see the graph on the right) does not mention the discontinuity. Show students this and comment appropriately. They will have the same experience with a graphics calculator.

On the web There is an interactive page on the reciprocal function and its graph on MathsNet at www.mathsnet.net/asa2/2004/c12recip.html.

Autograph displays the graph, despite the lack of definition at x = 0.



Example A suspension platform is hung from a ceiling by a number of identical parallel springs. When a weight is added, the extension in the system of springs is found to be inversely proportional to the number of springs. When there are 6 springs, the extension produced by 5 kg is 4 cm. Find the extension produced when there are 4 springs.

Model solution Let the number of springs be n and the extension be e cm.

1

When 6, 4, so244 24

6Hence, when 4, 6.

ke en nn e

k k en

n e

∝ ⇒ =

= =

= ⇒ = ⇒ =

= =

The extension is 6 cm.

Exercises Set questions on similar situations.

2 hours

Properties of graphs Identify and use interconnections between mathematical topics.

Understand the symmetry properties of functions, and when a function is even or odd.

Use physical contexts to plot and interpret graphs of functions, recognising when the functions are increasing, decreasing or stationary, including: • linear, quadratic and

cubic functions; • the reciprocal function y = k / x (x ≠ 0);

• the sine and cosine functions;

• the modulus function and a range of simple non-standard functions.

Odd, even and other types of function and graph

Class discussion Cubic and reciprocal graphs prompt discussion of issues of symmetry. Begin by defining the different types of symmetry of a graph y = f(x) and the words associated with them. These are: • odd functions, where f(–x) = –f(x); • even functions, where f(–x) = f(x). Illustrate with examples from the functions which students have met – parabolas, cubics and the reciprocals. Discuss how these correspond to geometric symmetries: • odd functions: the graph has rotation symmetry of order 2 about the origin; • even functions: the graph has reflection symmetry in the y-axis.

Introduce at least one new function. An example is the modulus function:

f(x) = | x | = when 0 when 0

x xx x

≥⎧⎨− <⎩

Discuss how to draw the graph of the function and highlight its principal features.

Note that the case of x = 0 may provoke some discussion: it can be included in the first line or the second of the definition. Introduce also, say,

f(x) = [ x ] (the integral part of x, so [ 3.6 ] = 3)

On the web This topic is explored further but accessibly at staff.imsa.edu/math/journal/volume4/articles/ ExploreEvenOdd.pdf. The PDF file is a good source of problems for more able students. It includes the result that any function can be written as the sum of an odd function and an even function.


Objectives Possible teaching activities Notes School resources Exercises

Give students examples to practise. Introduce further new functions as they work on exercises. Set questions that combine pencil and paper sketching with investigations on graphics calculators and using applets to extend their experience.

Further ideas to develop through these are: • sine and cosine functions over extended ranges, and their properties; • periodic functions; • increasing and decreasing functions; • first acquaintance with the functions y = ln x and y = ex.

Examples • Sketch the graphs of the functions y = | x – 1 | and y = | x – 1 | + | x – 3 |. • Sketch the graphs of the functions y = sin [ πx – π ] and y = | x |2 + 3 | x | + 2. • Find the domain over which y = x2 – 5x + 4 is an increasing function. • Solve the equation | x + 1 | = x2 – 5x + 4.

At the end of this activity, summarise the key points and definitions for future reference.

Use the applets from www.fi.uu.nl/wisweb/welcome_en.html:

Find the function Scope

Playing with functions

2 hours

Inverse functions

Find, graph and use the inverse function of those functions given by a one-to-one mapping or restricted to such mappings; know that the graph of the inverse function may be found by reflecting the graph of the function in the line y = x; solve a range of problems using inverse functions.

Use a range of strategies to solve problems, including working the problem backwards and then redirecting the logic forwards; set up and solve relevant equations and perform appropriate calculations and manipulations; [continued]

Finding an inverse function

Class discussion

Pose the simple problem of making t the subject of the formula 5 7 ,3tx += giving 3 7 .

5xt −=

Then ask students to reconsider the problem from the point of view of functions and graphs. The arrow diagram (top right) shows the functional relationship as a rule to map the domain elements {1, 4, 7, 10} to the range elements as shown. The same diagram drawn from right to left (or with the arrows reversed) represents the inverse function derived in a similar way.

The graph shows the two functions as 5 7 3 7, ;3 5x xy y+ −= = note that the letters have

changed but that the functions are unchanged. Ask students how the two graphs are related.

From these diagrams and the algebra explain what is meant by inverse function, establish how to derive an inverse and confirm that the two graphs are related by reflection in y = x. Now extend the concept of inverse to more complex situations. Return to the distinction between graphs which represent functions (or mappings) and those that do not. Find out if students remember how this is obvious from a graph. Convince them from this that an inverse (function) only exists if a function is one-to-one on its domain of definition. Explain how the domain can sometimes be restricted to make this so. (For example, y = x2 has an inverse if the domain is restricted to x ≥ 0.)

For more able students introduce the idea of inverse sine and inverse cosine functions, which depends on the idea of restricted domain.

1 → 4

4 → 9

7 → 14

10 → 19

5 7 3 7, ,

3 5x xy y y x+ −= = =



[continued] change the viewpoint or mathematical representation, and introduce numerical, algebraic, graphical, geometrical or statistical reasoning as necessary.

Exercises Set questions which cover: • revision of graphs to distinguish functions from relations; • determination of inverses of one-to-one functions; • sketches of functions and their inverses; • functions which can be considered one-to-one by suitably restricting their domains; • inverses of functions with restricted domains.

2 hours

Combinations of functions

Add, subtract and multiply two functions given in the form y1 = f1(x) and y2 = f2(x); write down, without simplification, the mathematical form for one function divided by another.

Understand the concept of a composite function and use the notation y = f(g(x)). Deconstruct a composite function into its constituent functions, using inverse functions.

Combining functions

Class discussion Explain that functions can be added, subtracted, divided and multiplied just like numbers. Give examples in the form of polynomials that are combinations of powers, and of brackets which multiply out to give equivalent expressions (identities).

Move on to the idea of composition. Take an example such as y = sin (x3 + 1). Ask students: • What would you do first to evaluate such a function at a given value of x?

List the steps:

3 3 31 sin( 1)1 sin( 1)

sin

x x x xp p p

q q

⎯⎯→ ⎯⎯→ + ⎯⎯→ +⎯⎯→ + ⎯⎯→ +

⎯⎯→

Each of the steps is a function in itself, and the end result is the composition of them all.

3f

g

h1

sin

x xp p

q q

⎯⎯→⎯⎯→ +

⎯⎯→

The function is finally expressed as y = h(g(f(x))).

Conversely, the deconstruction method allows the inverse to be constructed by reversing the steps. Thus:

1

1

11/ 3

h

g

f

arcsin1

y yr r

s s

−

−

−

←⎯⎯⎯− ←⎯⎯⎯

←⎯⎯⎯

That is: 1 1 11/ 3 f g h(arcsin 1) arcsin 1 arcsiny y y y

− − −− ←⎯⎯⎯ − ←⎯⎯⎯ ←⎯⎯⎯

So we have x = f–1(g–1(h–1(y))) = (arcsin y – 1)1/3.

On the web MathsNet has pages devoted to functions (www.mathsnet.net/asa2/2004/c3.html#1).

Another good source is at Visual Calculus (archives.math.utk.edu/visual.calculus/0/ compositions.5/index.html).



Exercises Give students exercises on these procedures.

Examples • If f(x) = x3 – 1, g(x) = x2 + x + 1 and h(x) = x – 1, find and simplify: f(x) + g(x); f(x) – g(x)h(x);

f( ) g( ) .f( ) h( )x xx x

+−

• If f(x) = x3 and g(x) = x + 1, find and simplify f(g(x)) and g(f(x)). Solve the equation g(f(x)) = f(g(x)).

Solving simultaneous equations with graphics calculators

Class discussion Students have considered simultaneous equations from an algebraic point of view and have related the algebra to the corresponding graphs. This section brings these ideas together and exploits the graphics calculator.

Begin by discussing an example. • Obtain a graphics calculator display of the curves: y = 5 + 6x – 3x2 – x3; y = x3 – 4x + 1. • Use the zoom function to obtain estimates (to one decimal place) of the x-coordinates of the

intersection points. • Hence solve the inequations: 5 + 6x – 3x2 – x3 > x3 – 4x + 1: 0 > 4 + 10x – 3x2 – 2x3.

2 hours

Using a graphics calculator to solve simultaneous equations

Identify and use interconnections between mathematical topics.


Use a graphics calculator, including use of the trace function, to show approximate solutions to physical problems requiring the location and physical interpretation of the intersection points of two or more graphs.

Exercises Set questions of the same sort as the example.

In addition, introduce (using the graphics calculator) the graphs of the exponential and logarithmic functions.

Use physical contexts (such as growth and decay situations) to create problems with the same type of question. The trace function can be used to relate the solutions to the graphs.

Using ICT Use the applet Growth from www.fi.uu.nl/wisweb/welcome_en.html.


Assessment


Which grows faster for x ≥ 0: the power function y = x3 or the exponential function y = ex? Justify your answer.

Look at these graphs.

a. One of the graphs shows the equation y = kx – x2, where k is a constant. Which graph is it?

b. One of the graphs shows the equation y = k / x, where k is a positive constant.Which graph is it?


The average speed for a fixed distance journey is inversely proportional to the time taken to complete the journey. A family travels in Europe by car. They travel exactly half their journey in 2 hours, then stop for lunch for 1 hour, and then take 3 hours over the second half of the journey.

How were the average speeds related on each part of the journey?

If the average speed for the first half of the journey was 72 kilometres per hour, what was the average speed for the whole journey?

A big wheel makes one complete revolution every 90 seconds. The wheel has a diameter of 20 metres. The bottom of the wheel is 2 metres above the ground. Two people get on the wheel and sit in a seat, and then the wheel starts to rotate. T seconds later their height above the ground is given by h = 2 + 8 sin 4T °.

Explain why this is an appropriate formula to use.

At what two consecutive times are the people 12 m above the ground?

A ship can only enter a harbour when the tide is in; it must have a minimum depth of water of 8 metres. The tide follows a daily sinusoidal variation given by the formula d = 5 sin 15t ° + 8, where t is the time in hours from midnight onwards, measured on the 24-hour clock.

At how many times in a day will the depth of water in the harbour be exactly 8 m? For how many hours a day can the ship enter the harbour?

Sketch how the level of the tide varies with the time of the day.

Find the inverse function of f(x) = 5x – 8.

Starting from the function y = x, describe how the function y = (5x – 3)2 is constructed.

Show how to deconstruct this function to return to the function y = x.

Unit 11A.8


GRADE 11A: Trigonometry 2

Functions, equations and identities

About this unit This is the second of two units on trigonometry for Grade 11 advanced. It builds on Unit 11A.4, Trigonometry 1. In this unit, students will develop their understanding of the relationship between Pythagoras’ theorem and trigonometry through equations and identities.





Previous learning To meet the expectations of this unit, students should already be able to solve right-angled triangles using Pythagoras’ theorem and trigonometric ratios. They should know the exact values for the sine, cosine and tangent of 0°, 30°, 45°, 60°, 90° and use these in relevant calculations. They should be able to use radians to calculate sector areas and arc lengths.

Expectations By the end of the unit, students will use mathematics to model and predict outcomes of real-world applications, and will compare and contrast two or more given models of a particular situation. They will use trigonometry to solve practical and theoretical problems. They will use Pythagoras’ theorem to show that sin2 θ + cos2 θ = 1 for any angle θ. They will plot the graphs of circular functions and solve simple problems modelled by these functions. They will generate mathematical proofs. They will approach problems systematically, knowing when to enumerate all outcomes and working to expected degrees of accuracy.

Students who progress further will in addition have more confidence in complex manipulation of equations and identities.

Resources The main resources needed for this unit are: • overhead projector (OHP) • Internet access, computer and data projector • dynamic geometry system (DGS) such as:


• computers with Internet access and dynamic geometry software for students


Key vocabulary and technical terms Students should understand, use and spell correctly: • sine (sin), cosine (cos), tangent (tan), secant (sec), cosecant (cosec),

cotangent (cot) • radian • equation, identity • symmetry, periodicity

UNIT 11A.9 7 hours






10A.1.2 Use mathematics to model and predict the outcomes of real-world applications.



11A.1.10 Approach a problem systematically, recognising when it is important to enumerate all outcomes.


10A.6.8 Use Pythagoras’ theorem to find the distance between two points, to solve triangles, to find Pythagorean triples, and to set up the Cartesian equation of a circle of radius r, centred at the point (α, β).

11A.8.6 Use the unit circle x2 + y2 = 1 to plot and describe the features of the graphs of the circular functions

f(θ) = sin θ, f(θ) = cos θ, f(θ) = tan θ,

where θ is measured in radians and 0 ≤ θ ≤ 2π; know the symmetries and periodicities of these functions; know that any point on the unit circle has coordinates (cos θ, sin θ), where θ is the angle the radius to the point makes with the positive x-axis.

2 hours

Extending the trigonometrical functions

1 hour

Trigonometrical identities

3 hours

More complex trigonometrical equations

1 hour

Radians and limits

10A.6.6 Derive and recall the exact values for the sine, cosine and tangent of 0°, 30°, 45°, 60°, 90° and use these in relevant calculations.

11A.8.7 Use Pythagoras’ theorem to show that sin2 θ° + cos2 θ° = 1 for any angle θ°.

12AS.7.4

10A.7.2 Use radians to calculate sector areas and arc lengths.

Know the polar form of a complex number, using the modulus r and the argument θ, and the results that x = r cos θ and y = r sin θ and r2 = x2 + y2 = zz*.

11A.6.1 Understand the concept of a limiting value.

10A.6.5 Know the standard trigonometric ratios, and their standard abbreviations, for sine of θ, cosine of θ and tangent of θ, given an angle θ in a right-angled triangle, and use these ratios to find the remaining sides of a right-angled triangle given one side and one angle or to find the angles given two sides.

11A.8.8 Solve simple problems modelled by circular functions.

Unit 11A.9


Activities


Working with trigonometrical functions

Class discussion Students have previously extended the definition of sine and cosine to cope with the sine and cosine rules in acute- and obtuse-angled triangles. They will have experienced the extended domain of definition for the sine and cosine waves from their graphics calculators.

The diagram shows the general definition of the circular functions for an angle between 0° and 90°. The coordinates of the point on the circle shown are found by using elementary trigonometry in a right-angled triangle with unit hypotenuse. Explain to students how any angle (of any size and sign) can be shown on such a diagram and that the coordinates of the point obtained are taken as the definition of sine and cosine for angles outside the range 0° to 90°. Point out that this is consistent with the way ratios for obtuse angles were found when solving triangles using the sine and cosine rules and the area formula.

Use reflection symmetries to show that consistency. Express the results obtained by considering signs in the forms sin (180° – θ) = sin θ and cos (180° – θ) = –cos θ.

Ask students to consider the corresponding problem for the tangent function. • How should this be generalised?

Students may respond in terms of coordinates on the diagram or by using the relation sintancos

= θθθ

. Either can be taken as an acceptable definition, but show that they are

equivalent.


2 hours

Extending the trigonometrical functions

Work to expected degrees of accuracy, and know when an exact solution is appropriate.

Use the unit circle x2 + y2 = 1 to plot and describe the features of the graphs of the circular functions f(θ) = sin θ, f(θ) = cos θ, f(θ) = tan θ, where θ is measured in radians and 0 ≤ θ ≤ 2π; know the symmetries and periodicities of these functions; know that any point on the unit circle has coordinates (cos θ, sin θ), where θ is the angle the radius to the point makes with the positive x-axis.

Use Pythagoras’ theorem to show that sin2 θ° + cos2 θ° = 1 for any angle θ°.

Exercise Take time now to practise these ideas. Set questions such as the following. • Find an acute angle θ such that sin 460° = ± sin θ; do this first by studying the diagram of a

unit circle on coordinate axes, then by studying the graph produced by your calculator. • Without using a calculator, use the fact that sin 30° = 0.5 to obtain the values of sin 150°,

sin 210°, sin 330°, sin 390°. • Without using a calculator, decide which of these are equal: sin 50°, sin 130°, sin 310°, …

• Write down all the angles in the range 0° < θ < 720° for which tan θ = 1

3.

Emphasise the importance of drawing diagrams to make a convincing case for the answers.

Summarise the outcomes of all this by deriving the ACTS rule for the signs of the functions (see the diagram on the right).

T (for example) signifies that only the tangent is positive for an angle whose defining radius lies in that quadrant. In the quadrant marked A, sine, cosine and tangent are all positive.

Unit 11A.9



Some special cases and a new look at Pythagoras

Class discussion Review the special cases of 30°, 45° and 60°, and derive the values of sine, cosine and tangent for these angles. Stress that students should commit the values to memory, but that the best way of doing so is repeatedly to draw the diagrams in which they occur.

sine cosine tangent

30° 12

32

13

45° 12

12

1

60° 3

2

12

3

Go on to derive the trigonometrical equivalent of Pythagoras’ theorem by writing it down from the defining circle in the notes, and showing that this always works without regard to the quadrant where the defining radius occurs.

cos2 θ + sin2 θ = 1 can be obtained from the unit circle diagram but also by using a simple right-angled triangle.

2 2 2

2 2

2 2

2 2

2 2 2

2 2

2 2

2 2

1

cos sin 1and

1

1 tan sec

p q rp qr r

p q rq rp p

+ =

⇒ + =

⇒ + =

+ =

⇒ + =

⇒ + =

θ θ

θ θ

The corresponding result for cosec2 θ is obtained similarly. Define these new ratios at this point. Mention that cosec2 θ (say) is conventional shorthand for (cosec θ)2.

Exercises Now set questions which pull all these points together.

Examples • If sin θ = 8⁄17 and θ is obtuse, find without using a calculator the exact values of the other five

trigonometric ratios for that angle.

• If tan θ = 12⁄5, find by exact arithmetic the possible values of sin θ and cos θ.



1 hour

Trigonometrical identities


Use Pythagoras’ theorem to show that sin2 θ° + cos2 θ° = 1 for any angle θ°.

Identities

Class discussion Review the distinction between equations and identities. Give students an example of an identity and ask for suggestions on how to start. For example: • Prove that (1 – cos2 θ)2 + (1 – sin2 θ)2 = cos4 θ + sin4 θ.

Respond to the discussion by developing the solution; at all stages pause to invite contributions.

Emphasise the importance of starting with one side only and working down the page until the other side is reached. Do not accept solutions with the proposition stated and the appearance of deductions from it; if the proposition to be proved is written down, it must be stated that it is to be proved.

Model solution 2 2 2 2

2 4 2 4

2 2 4 4

4 4

LHS (1 cos ) (1 sin )1 2cos cos 1 2sin sin2 2(cos sin ) cos sincos sinRHS

QED

= − + −= − + + − += − + + += +=

θ θθ θ θ θθ θ θ θ

θ θ

QED stands for the Latin phrase quod erat demonstrandum; it means ‘that which was to be shown’ and by convention is used to round off a proof.

Exercises Give students time to come to terms with the challenge of this topic. They will require plenty of questions to establish that: • they can use the identities with facility; • they have memorised the exact values of the ratios for 30°, 45° and 60°; • they are willing to present solutions correctly.

When identities reach an appropriate level of complexity, demonstrate a case where it is necessary to start with each side separately to reach common ground.

On the web There is plenty on this topic and more at MathsNet, starting for example at www.mathsnet.net/asa2/2004/c2.html#4.



3 hours

More complex trigonometrical equations Use mathematics to model and predict the outcomes of real-world applications; compare and contrast two or more given models of a particular situation.

Solve simple problems modelled by circular functions.

Equations

Class discussion Begin with a simple case such as: • Find all angles in the range 0° < θ < 360° for which sin θ = 0.4.

Ask students to solve this. Look for an awareness in the responses that: • there will be more than one answer; • the calculator is essential to find even one.

Draw a good diagram of the graph y = sin θ to illustrate how all (in this case just both) answers can be found by using the calculator once and studying the periodic nature of the graph. Make clear that drawing diagrams or graphs in a case like this helps to substantiate the answer.

Next consider a more elaborate case such as this one: • Find all angles in the range 0° < θ < 360° for which cos 2θ = 0.7.

Give students time to think about this, and the importance of keeping all the solutions. Allow students to pursue any faulty reasoning to the full in order to demonstrate their errors.

Consider an example of this type, where a quadratic results: • Find all angles in the range 0° < θ < 360° for which 4 cos2 θ – 3 cos θ = 1.

Give students time to realise that the two possible solutions of the quadratic each give rise to more than one solution.

Finally consider practical applications, such as tides and other oscillations, where the sine and cosine waves model variation of distance and time. Here is an example. • The depth of water in a harbour is modelled by d = b + a sin ωt, where t hours is the time

elapsed and d metres is the depth. In the case that a = 5.6, b = 7.3 and ω = π⁄6, find for how long the depth is more than 10 metres.

Exercises

Set questions which explore all the ground covered. These should include: • simple questions, such as finding solutions in a given range to tan x° = 3; • equations which involve multiple angles; • cases which involve solving a quadratic or cubic; • cases in which the problem is expressed in terms of sec or cosec and which require an

intermediate transformation into sin or cos; • problems in which there is a mixture of sin and cos (or of tan and sec, say) for which a use of

Pythagoras’ theorem is needed as a first step; • cases which have no solutions (perhaps because, say, sin x° = 2); • cases where an exact answer is possible and expected.

More able students may appreciate the formulae for the general solution of trigonometric equations:

sin x° = sin y° ⇒ x = 180n + (–1)n y, where n ∈

and its parallels.



1 hour

Radians and limits

Use radians to calculate sector areas and arc lengths.

Work to expected degrees of accuracy, and know when an exact solution is appropriate.

Understand the concept of a limiting value.

A return to radians

Class discussion

Ask students to explain radian measure. Use the response to re-establish the two key results, c = r θ and A = 1⁄2 r2 θ.

Take a moment to review the angles for which exact values of sine, cosine and tangent can be calculated, and express these in their radian terms.

Ask students if they know why radian measure is useful.

Move on to the important limit result. Present this as an unfolding argument, so that students are able to follow it and still grasp the final result with clarity.

2 2 21 1 12 2 2

0

0

area of area of area oftriangle OAB the sector triangle OAC

tan sintan sinsin sincoscos 1 1sin sin

sincos 1

sin1 lim 1

sinWe conclude that lim 1.

r r r

→

→

> >

⇒ > >⇒ > >

⇒ > >

⇒ < <

⇒ < <

⇒ ≤ ≤

=

θ

θ

θ θ θθ θ θθ θ θθθθ θ θ

θθθ

θθ

θθ

Allow time for students to question the last few steps; this applies particularly to the way the strict inequations give way progressively to equality in the last step. Draw accompanying diagrams to make sure that the result is clear.

Explain that this justifies the physicists’ use of radian measure and the approximation sin θ = θ. Add that the result is an essential first step towards finding a gradient of a trigonometrical graph.

Explain how to use this result to evaluate limits such as that of sin2θθ

as θ → 0.

sine cosine tangent

30° 6π 1

2 3

2

13

45° 4π

12

12

1

60° 3π 3

2

12

3

The sector of the circle here has centre O, and the radius is r. AB is a tangent, so ∠OAB is a right angle.

Exercises This section has been mainly theoretical, laying the foundations for results used elsewhere, whether in physics or mathematics. Nevertheless, set questions to test the understanding of the limit, such as the one just given.


Assessment


Explain why sin (2π – θ) = sin θ.

Give the exact value of cos 5⁄4 π. What other angle has the same cosine value?

Draw the tangent line at the origin for the curve y = sin x. Use this to estimate the value of sin x for small values of x. Explain why it is important to use radian measure in this context.

Use Pythagoras’ theorem to show that sin2 θ° + cos2 θ° = 1 for any angle θ°. Verify this result for the angles 30°, 45° and 60°. What happens when θ° = 90°?

A point P moves in the plane so that at time t its x-coordinate is 2 cos t and its y-coordinate is 2 sin t. By eliminating t between those two equations find the equation of the curve traced by point P. Determine the times in the range 0 ≤ t < π when the x-coordinate has the value 1, and find the exact values of the y-coordinate at those times. For what values of t are the x- and y-coordinates equal?


An angle x° is obtuse and its sine is 7⁄25. Find exact values (without using a calculator, and showing all working) for cos x° and cot x°.

Find all solutions x for each of the following equations on the interval 0° ≤ x ≤ 360°.

a. sin x = 32

−

b. cos 2x = –1

c. tan 3x = 12

d. sec 4x = 2

Evaluate the limits (a)

0

sin2lim ;3x

xx→

⎛ ⎞⎜ ⎟⎝ ⎠

(b) 2

20

3lim1 cosx

xx→

⎛ ⎞⎜ ⎟−⎝ ⎠

(where x is measured in radians).

A sector of a circle has area 40 cm2 and radius 5 cm. Find its perimeter.

Prove that (a) 4 21 tan 2 sec ;− = −θ θ (b) 21 cosec cot cosec .

1 cos= +

−θ θ θ

θ

Unit 11A.9


GRADE 11A: Probability 2

Risk, trends and simulation

About this unit This is the second of two units on probability for Grade 11 advanced. It builds on Unit 11A.5, Probability 1, and introduces students to models and simulations. It also gives them the opportunity to carry out research on risk.





Previous learning To meet the expectations of this unit, students should already be able to use relative frequency to estimate probability. They should be able to calculate theoretical probabilities for outcomes of independent events, and to compare theoretical and experimental probabilities. They should be able to list systematically all the possible outcomes of an experiment. They should know that if A and B are mutually exclusive, the probability of A or B is the sum of the probabilities of A and of B.

Expectations By the end of the unit, students will understand risk as the probability of the occurrence of an adverse event. They will use simple simulations and consider trends over time using a moving average. They will analyse results to draw conclusions, and will use a range of graphs, charts and tables to present their findings. They will use relevant statistical functions on a calculator and ICT applications to present statistical tables and graphs.

Students who progress further will recognise and use the underlying process of modelling to further their understanding.

Resources The main resources needed for this unit are: • overhead projector (OHP) • Internet access, computer and data projector • spreadsheet software such as Microsoft Excel • computers with Internet access and spreadsheet software for students • graphics calculators for students

Key vocabulary and technical terms Students should understand, use and spell correctly: • time series, historigram, moving average • trend, cyclical, seasonal and residual variation • simulation • risk

UNIT 11A.10 10 hours






11A.1.1 Solve routine and non-routine problems in a range of mathematical and other contexts, including open-ended and closed problems.



12AQ.1.2 Use statistical techniques to model and predict the outcomes of statistical situations, including real-world applications; work to definitions and perform appropriate tests.

11A.1.12 Synthesise, present, discuss, interpret and criticise mathematical information presented in various mathematical forms.

12AS.1.2 Use mathematics to model and predict the outcomes of substantial real-world applications, and to compare and contrast two or more given models of a particular situation.

10A.8.11 11A.12.4 Understand risk as the probability of occurrence of an adverse event; investigate some instances of risk in everyday situations, including in insurance and in medical and genetic matters.

11A.13.1 Consider trends over time and calculate a moving average.

Make inferences and draw conclusions from the formulation of a problem to the collection and analysis of data in a range of situations; select statistics and a range of charts, graphs and tables to present findings.

11A.14.1 Use coins, dice or random numbers to generate models of random data.

2 hours

Trends over time

4 hours

Simulation

4 hours

Risk

11A.15.1 Use a calculator with statistical functions to aid the analysis of large data sets, and ICT packages to present statistical tables and graphs.

Unit 11A.10


Activities


2 hours

Trends over time Consider trends over time and calculate a moving average.

Use mathematics to model and predict the outcomes of real-world applications; compare and contrast two or more given models of a particular situation.

Synthesise, present, discuss, interpret and criticise mathematical information presented in various mathematical forms.

Time series

Class discussion Present the class with (say) sales data of a particular item at a business by year. Explain that this is an example of a time series: a series of observations of a variable taken at intervals in time. Ask students how they would represent this: obvious answers are graphs, bar charts, histograms.

Focus on graphical representation – sales plotted against time, for example. Say that this is sometimes called a historigram (not the same as histogram).

Explain the terms: • trend: the basic underlying movement with other fluctuations smoothed out; • seasonal variations: short-term regular fluctuations about the trend, such as may be evident

in any particular year’s breakdown by months; • cyclical variations: long-term fluctuations about the trend, corresponding to highs and lows in

economic activity (evident in the graph on the right where there is a clear five-year cycle); • residual variation, which is caused by unusual events such as a serious fire in business

premises.

Explain how to estimate trend: • the simplest technique is a line of best fit, which is applicable when there are no cyclical or

seasonal variations; • the method of moving averages is available when there are cyclical variations.

Explain how to calculate a moving average by estimating the length of a cycle (five years in the graph in the notes) and working out an average for each consecutive set of five figures. Index these averages by the mid-value of the series of five, as in the second graph. The moving averages now are the basis of a trend graph. Explain that a line of best fit can be used to extend the graph into the future to form the basis of prediction.

Point out to students that different styles of line are important to distinguish the different models here, to distinguish the trend from the historical, and to distinguish predictions from data values.

Go on to discuss different ways of looking at such data. When considering sales, for example, the model to use is: sales = trend + seasonal variation + cyclical variation + residual variation

This will work when the trend is changing slowly, or over the short term.

Examples of time series data: • birth dates (by week); • unemployment/employment figures (monthly); • share index numbers (published daily for the

international stock markets).

A typical sales graph indexed by years

The same graph with a moving-average graph added


Unit 11A.10



Moving averages using a spreadsheet In the extract from a spreadsheet on the right, based on Microsoft Excel, the first column gives the year and quarter for which the sales appear in the second column.

Excel’s chart drawing facility gives the chart below, and it has added a moving average based on a period of four quarters. (The moving average is plotted over the right-hand end-point of the period by Excel rather than the conventional mid-point.) Explain this to students so that they can use it as a tool.

Exercises Set questions which present data to be plotted: • as a historigram; • as a moving average (on the same graph).

Ask students to make forecasts on the basis of their estimates of the line of best fit to the moving-average figures.

In addition, set questions that expect students to be critical about the value of this process in particular cases.

For an example, see the project outlined on the right.

Project For a period of four weeks: • note the exchange rate between two major

world currencies (e.g. US dollars and pounds sterling);

• plot a line graph of the data; • estimate the trend and predict the exchange

rate for each day in the following week; • compare those estimates with the actual values

when they become available.



4 hours

Simulation Use coins, dice or random numbers to generate models of random data.

Use a calculator with statistical functions to aid the analysis of large data sets, and ICT packages to present statistical tables and graphs.

Class discussion Ask students about their expectations when tossing a coin. Various ideas should come into play: • it is impossible to predict the next toss; • there is a probability model of a fair coin; • to verify any probability value requires a large number of tosses.

Explain to students that the models help in considering long-term trends (like moving averages do) but that the sort of variation which occurs in practice can be experienced only by doing experiments.

Explain that a simulation assumes that the underlying model is correct. It is useful to generate some data to represent the variation that would happen in practice.

Ask these questions: • In tossing 100 coins the number of heads is expected to be 50. In practice, how often would

you expect to get, say, 55? • Experience may show that the average queue in a doctor’s surgery is 6 patients. How often

are there 15 people waiting? What implications are there for the accommodation needed in the waiting area?

Explain the basis of simulation using random numbers by an example. Explain to students what a table of random numbers is, and agree a probability distribution for coin tossing (that the probability of head is 0.5, and tail 0.5, for example).

Explain how to allocate random digits to simulate those, by considering 0, 1, 2, 3 and 4 to be occurrences of head, and the rest of the digits tails.

Get students to do a small-scale experiment using the random-number keys of their calculators.

Explain that coins and dice can be used to simulate too.

Experiment Once the idea is clear, and also the implications in terms of data management, move on to using a computer to simulate the experiment. Alternatively, get students to work in teams to generate sufficient data. Require students to present their data using the techniques from earlier units and to summarise their conclusions.

In this sort of work, get students first to be clear about what questions they wish to ask, and to take account of those questions when they are designing their experiments.



Simulation using a spreadsheet such as Excel The extract from a spreadsheet in the notes on the right shows one simulation of a coin-tossing experiment. • The first column is the index number of the throw; there are 22 of these but in practice a

much larger number should be used. • The second column simulates the throw of a coin: 0 denotes tail and 1 head. This is done by

B2=INT(0.5+RAND()), and replication down the column. • The first pivot table below shows an analysis of the results: there were 11 heads in this

experiment. • The third column counts the run length (the number of consecutive heads or tails), and the

fourth column records the run lengths that occur. This is done by entering 1 into C2 and C3=IF(B3=B2,C2+1,1) with replication down to C23.

• The fourth column records the runs by setting D2=IF(C3>C2,,C2) with replication down to D22. The last cell D23 is blank since without C24 it is impossible to tell whether the current run has finished. Ignore the zeros in column D.

• The second pivot table records the results. Sufficient data should provoke some interesting questions. Ask students to pose some, for example: • What is the distribution of run lengths? • What is the waiting time to (say) the fifth head?



4 hours

Risk Understand risk as the probability of occurrence of an adverse event; investigate some instances of risk in everyday situations, including in insurance and in medical and genetic matters.

Solve routine and non-routine problems in a range of mathematical and other contexts, including open-ended and closed problems.

Use a calculator with statistical functions to aid the analysis of large data sets, and ICT packages to present statistical tables and graphs.

Statistical project

Class discussion Discuss with students their understanding of the term risk (the probability of the occurrence of an adverse event). This may be applied to travel, for example, where students may have heard that air travel is the ‘safest’ mode. Ask students what such a statement may mean, and whether they consider air travel to be risky. Ask whether they think it is less risky to travel by road.

Try to establish some sort of agreed approach to the questions.

The website www.bast.de/htdocs/fachthemen/irtad/english/englisch.html is a useful source. This contains an overview of international road traffic and accident data and includes calculated risk values for the year 2002.

The World Health Organisation publishes an annual report with statistical annexes, which are offered for downloading as Excel files (see www.who.int/whr/en).

Project Ask students to consider a particular area of risk of their choice, for example: • travel mode risks; • health risks – be specific; • accident risks in sport – such as sprains in tennis. Ask students to research their area of risk and to provide a report. Before they start, check that students have: • provided a clear question to research; • identified a data source to work on.

Stress to students the importance of sourcing, editing and summarising their data carefully, and of making their conclusions clear.

On the web There are statistics available on a variety of subjects (for example, on the UK National Statistics site at www.statistics.gov.uk).

Note that for classroom use many sites offer only PDF files, so it is important to find sites that offer Excel files of data. An example of what is available from this site is www.statistics.gov.uk/ STATBASE/ssdataset.asp?vlnk=8397.

Also try the United Nations website cyberschoolbus.un.org/infonation3/menu/ advanced.asp. This allows data from groupings of countries to be selected and compared (e.g. from North Africa and the Middle East).

More data sources are listed on the Oundle-TSM site at www.tsm-resources.com/mlink.html#stats.


Assessment


The following table shows the sales of a product during the years 1989–1997.

Year 1989 1990 1991 1992 1993 1994 1995 1996 1997

Sales (000s)

448 365 465 466 392 483 493 413 510

a. Draw a historigram of these data.

b. Comment briefly on the types of variation present in this time series.

c. Calculate moving averages over an appropriate period and plot these on your graph. Draw a trend line by eye.

d. Estimate the trend value for 1997 and hence estimate the actual sales in 1998.

Assessment Set up activities that allow students to demonstrate what they have learned in this unit. The activities can be provided informally or formally during and at the end of the unit, or for homework. They can be selected from the teaching activities or can be new experiences. Choose tasks and questions from the examples on the right to incorporate in the activities. a. Explain how to use random numbers to simulate throws of a fair dice.

b. Use random numbers generated by your calculator to generate a sample of size 21 in accordance with your simulation design.

c. Calculate the average score of the sample.

d. On the basis of your sample, calculate the relative frequency of a throw that is immediately followed by another of the same score.

e. Do you think that your two calculated values are in any sense representative?

Unit 11A.10

225 | Qatar mathematics scheme of work | Grade 11 advanced | Unit 11A.11 | Calculus © Education Institute 2005

GRADE 11A: Calculus

Introduction to differentiation

About this unit This is the only unit on calculus for Grade 11 advanced.





Previous learning To meet the expectations of this unit, students should already be able to draw the tangent line at a point on the graph of a function, calculate the slope of this line and state whether the function is increasing, decreasing or stationary at the point.

Expectations By the end of the unit, students will develop and explain chains of logical reasoning, using correct mathematical notation and terms. They will generate mathematical proofs and generalise when possible. They will understand the concept of a limit and find derivatives of functions.

Students who progress further will understand the concept of gradient and its connection with differentiation.




Key vocabulary and technical terms Students should understand, use and spell correctly: • limit, continuity, discontinuity • defined, undefined, continuous, differentiable • derivative, derived function, differentiation, tangent, normal, gradient • displacement, velocity, acceleration










12AS.1.6


11A.1.9 Generalise whenever possible.


10A.5.15 Draw the tangent line at a point on the graph of a function, calculate the slope of this line and interpret the behaviour of the function at that point, knowing whether the function is increasing or decreasing at the point, or stationary.

11A.6.1 Understand the concept of a limiting value; e.g. explain that: • the sequence of terms 1, 4, 9, 25, …, n2, … gets larger as n

gets larger and diverges as n tends to infinity; • the sequence of terms 31 2 4

2 3 4 5 1, , , , ..., , ...nn+ gets closer to 1 as n

gets larger and converges to 1, the limit of the sequence, as n tends to ∞;

• the function f(x) = 1 / x tends to –∞ as x tends to zero from the negative side, but tends to +∞ as x tends to zero from the positive side. At x = 0 the function is not defined. When x tends to +∞ the function tends to zero from the positive side. When x tends to –∞ the function tends to zero from the negative side. The lines x = 0 and y = 0 are asymptotes of the function;

• the function f(x) = x tends to 2 when x tends to 2 from just below 2 or from just above 2. For this function we can write

2lim f( ) 2x

x→

= .

11A.6.2 Consider a chord across the graph of the function y = f(x) between the points with coordinates (x, f(x)) and (x + h, f(x + h)), and show that, if θ is the angle that this chord makes with the positive x-axis,

then f( ) f( ) f( ) f( )tan .

( )x h x x h xx h x h

θ + − + −= =+ −

1 hour

Differentiation 1

1 hour

Differentiation 2

2 hours

Differentiation 3

2 hours

Limits

2 hours

Differentiation 4

2 hours

Differentiation and problem solving

2 hours

Rates of change and problem solving

11A.6.3 Evaluate:

0

f( ) f( )limh

x h xh→

+ − for the following functions:

f(x) = x2, f(x) = x3, f(x) = x–1.

12AQ.7.1 Know that the derivative of f′(x) is called the second derivative of the function y = f(x) and that this can also be written

in the forms f′′(x) or 2

2dd

yx

;

know that higher derivatives may be taken in the same way.

Unit 11A.11





11A.6.4 Understand that the limit

0

f( ) f( )limh

x h xh→

+ − for the function y = f(x) is represented

geometrically as the slope of the tangent to the curve for this function at the point (x, f(x)).

12AQ.7.2

10A.7.4 Solve problems involving compound

measures, including average speed, population density, density, pressure and power.

11A.7.1 Identify the tangent at the point (x, f(x)) on the function y = f(x) as the derivative of the function at this point, and denote the

derivative by either of the two common notations dd

yx

and f′(x);

interpret the derivative as a rate of change.

Interpret the numerical value of the derivative at a point on the curve of the function; know that: • when the derivative is positive the

function is increasing at the point; • when the derivative is negative the

function is decreasing at the point; • when the derivative is zero the function is

stationary at the point.

11A.7.2 Calculate derivatives of simple powers of x from first principles. 11A.7.3 Know the general result that if f(x) = axn, where a is constant, then

f′(x) = anxn–1 for all real values of n.

11A.7.4 Know that if f(x) = axn ± bxm, where a and b are constants, then f′(x) = anxn–1 ± bmxm–1.

12AQ.7.8 Understand that given any function f(x) = f1(x) + f2(x) then the derivative of this sum of two functions is f′(x) = f1′(x) + f2′(x).


Activities


1 hour

Differentiation 1

Develop chains of logical reasoning, using correct mathematical notation and terms.


Consider a chord across the graph of the function y = f(x) between the points with coordinates (x, f(x)) and (x + h, f(x + h)), and show that, if θ is the angle that this chord makes with the positive x-axis, then

f( ) f( )tan( )

f( ) f( ) .

x h xx h x

x h xh

θ + −=+ −

+ −=

Calculating gradients of curves

Class discussion Start differentiation intuitively as well as formally, to establish concepts at the same time as formal calculation. Encourage students to relate their calculations to clearly drawn diagrams at all times.

Present students with the sketch of the parabola y = x2 between (–3, 9) and (3, 9). Ask if the curve is steeper at Q than at P. Remind students if necessary that the gradient of a curve is the gradient of a tangent to the curve at that point.

Ask students how they would calculate the gradient of the curve at (say) (2, 4). Use the responses to bring out the idea of an approximating chord for the tangent at the point.

First point Second point Gradient of the connecting chord

(2, 4) (3, 9) 5 (= 4 + 1)

(2, 4) (2.1, 4.41) 4.1

(2, 4) (2.01, 4.0401) 4.01

(2, 4) (2.001, 4.004 001) 4.001

... … …

Students will easily see the pattern here. Get them to do the process graphically on a calculator using the zoom function. Draw out from them three related conclusions: • the smaller the x-step, the straighter the graph appears; • the calculated gradient approaches 4; • the gradient of the tangent must be 4.


Exercises

Have students repeat this approach for themselves by completing a similar table and using it to calculate the gradient of the tangent at (1, 1) and similarly at (3, 9). Some pupils will succeed, many will struggle, but all will benefit from trying this.

Class discussion Repeat the argument with one of the new cases. Make sure that students can suggest all steps for themselves. List the cases in a table for use in the next step.

Unit 11A.11



1 hour

Differentiation 2

Generalise whenever possible.

Identify the tangent at the point (x, f(x)) on the function y = f(x) as the derivative of the function at this point, and denote the derivative by either of the two common

notations dd

yx

and f′(x);


Calculate derivatives of simple powers of x from first principles.

Know the general result that if f(x) = axn, where a is constant, then f′(x) = anxn–1 for all real values of n.

Know that if f(x) = axn ± bxm, where a and b are constants, then f′(x) = anxn–1 ± bmxm–1.

Class discussion Show students the results obtained in the previous section. Ask them to point out the pattern.

Point Gradient

(1, 1) 2

(2, 4) 4

(3, 9) 6

Ask students to give the value of the gradient at (4, 16); it will be 8. Ask them also to use the pattern to give the gradient at (x, x2); it is 2x.

Now move on to the general case.

The next table shows one way of doing this. Get students to do the calculations themselves or accept them from you according to the degree of confidence they are showing in the process. Get them to see the pattern and anticipate the result.

Point Second point Gradient

(x, x2) (x + 1, (x + 1)2) 2x + x2

(x, x2) (x + 0.1, (x + 0.1)2) 2x + 0.1x2

(x, x2) (x + 0.01, (x + 0.01)2) 2x + 0.01x2

Class exercise and subsequent discussion Now have students work the argument for themselves on slightly more complex cases such as: • 2x2 • 3x • 2x2 + 3x + 4 • 2x3

Share these problems among the class so that more cases are covered and so that students have chance to discuss them. The last case here brings in revision of Pascal’s triangle and the binomial theorem. Ask for more conjectured generalisation.

Class discussion Summarise this with the rule for axn and the rule for sums of such functions.

On the web There is a comprehensive coverage of differentiation from beginners’ level upwards at MathsNet. One page to consult for this unit is www.mathsnet.net/asa2/2004/c1.html#5.

Differentiation of polynomials has an interactive page at Waldo’s Interactive Maths, www.waldomaths.com; follow the prompts 16–19→Calculus→Differentiating polynomials.



2 hours

Differentiation 3

Evaluate 0

f( ) f( )limh

x h xh→

+ −

for the following functions:

f(x) = x2, f(x) = x3, f(x) = x–1.

Formal limits

Class discussion Remind students of function notation. Repeat the argument of the last section but this time in abstract terms for a general functional graph of y = f(x). Use it to reach the definition of a limit in functional notation. Use the limit definition for the case y = x2, and emphasise that although h can be arbitrarily small it is never actually zero because: • that would mean there was only one point and so no chord; • it would entail dividing by zero in the algebra.

Show the class the two different notations in use: • f′(x)

• dd

yx

and explain the way they are used.

0

f( ) f( )f ( ) limh

x h xxh→

+ −′ = by definition

In the case of y = x2, we have: 2 2

0

2 2 2

0

2

0

0

0

d ( )limd

2lim

2lim

(2 )lim

lim(2 )

2

h

h

h

h

h

y x h xx h

x xh h xh

xh hh

h x hh

x h

x

→

→

→

→

→

+ −=

+ + −=

+=

+=

= +

=

Exercises Set simple questions which explore this new notation, and check that students are using it correctly. Examples to use are: • 2x2 (just like the example students have seen); • x (to complete the situation); • 5x – 3x2 (to show the way a combined expression works); • x3 (to show a more complicated case, using the binomial theorem);

• 1x

(difficult).

The last set of questions will enable students to discover from experience some key ways that the differentiation process works. Check that they have managed at least to start the case of x3; allow them to continue as far as they can without intervention.



Class discussion Remind students of three problems that they have considered in recent units; • the infinite geometric series, which may or may not converge;

• the graph of 1;yx

=

• the limit 0

f( ) f( )lim .h

x h xh→

+ −

Explain to students that these problems all concern ‘difficult’ values. These are (corresponding to the above cases): • infinity (which is never reached); • a value where the function is undefined (x = 0); • a value where the function is undefined (h = 0).

Ask students to give some more examples of this kind of problem.

Now ask students to put their calculators into radian measure and to generate a graph of the

function sin xx

. Ask them to copy the sketch on a piece of paper or on the board or OHT.

Ask students to consider the problem again and to look for ‘difficult’ value(s). Use the discussion to draw students’ attention to the facts that: • the function appears to be defined at x = 0 from the graphics calculator display; • the function is certainly undefined at x = 0.

Use this example to demonstrate the concept of limit as a value that is approached but not

necessarily attained. Contrast this with 1yx

= , where the undefined value corresponds to an

infinity in the graph. Finally point out that a function can only be described as continuous at x = a (say) if: • it is defined there; • its limiting value from both sides is the value of the function at the point.

On the web There is a page on removable discontinuities at http://tutorial.math.lamar.edu/AllBrowsers/2413/Continuity.asp.

2cos 1xy

x−= according to Autograph

Although the function is undefined at x = 0, the graph appears to be continuous there. It is an example of a function whose definition can be supplemented by y = –1⁄2 at x = 0 to make it well defined for all x and also continuous for all x.

2 hours

Limits Understand the concept of a limiting value; e.g. explain that: • the sequence of terms 1, 4,

9, 25, …, n2, … gets larger as n gets larger and diverges as n tends to infinity;

• the sequence of terms 31 2 4

2 3 4 5 1, , , , ..., , ...nn+ gets

closer to 1 as n gets larger and converges to 1, the limit of the sequence, as n tends to ∞;

• the function f(x) = 1/x tends to –∞ as x tends to zero from the negative side, but tends to +∞ as x tends to zero from the positive side. At x = 0 the function is not defined. When x tends to +∞ the function tends to zero from the positive side. When x tends to –∞ the function tends to zero from the negative side. The lines x = 0 and y = 0 are asymptotes of the function;

• the function f(x) = x tends to 2 when x tends to 2 from just below 2 or from just above 2. For this function we can write

2lim f( ) 2x

x→

= .

Exercises

Set questions which require students to look critically at the domains or definitions of functions and to compare these with limits obtained by algebra or from inspection of a calculator display.

Examples

• Graph 2cos 1x

x− . Is the function undefined for any real number? Does the function have a

right-hand and/or left-hand limit at such point(s)? Can the function be supplemented by a defined value at any point(s) where it is undefined so that it will then be continuous?

• Consider the function 2 7 12

4x x

x− +

−. Ask the same questions.



2 hours

Differentiation 4 Know the general result that if f(x) = axn, where a is constant, then f′(x) = anxn–1 for all real values of n.

Know that if f(x) = axn ± bxm, where a and b are constants, then f′(x) = anxn–1 ± bmxm–1.

Identify the tangent at the point (x, f(x)) on the function y = f(x) as the derivative of the function at this point, and denote the derivative by either of the two common

notations dd

yx

and f′(x);


Class discussion Begin by asking students what results they remember from the previous exercises. For example: • What is the derivative of x2, of 2x2, of 3x – 2x2?

• What is the derivative of 1?x

Use the responses to decide how much of this needs to be worked again (for less confident students) and how much has shown the underlying patterns (for more able students). Expect to have to work the case of 1 / x since the algebra is difficult but also because it shows how the abstract definition can be used without the need for a diagram.

Summarise the rules that emerge from all this:

• 1dd

n nyy x nxx

−= ⇒ =

• df( ) f ( )d

yy a x a xx

′= ⇒ =

• df( ) g( ) f ( ) g ( )d

yy x x x xx

′ ′= + ⇒ = +

Take care to judge the level of abstraction according to students’ ability. In all cases, illustrate the abstract rule with explicit examples, so that the simplicity of the technique is clear. Make sure that students realise that the index rules of algebra play an important role.

Exercises Now that the rule has become clear, formalise the procedure. Set plenty of questions to cover this ground.

Examples • Find the derived function in each of the following cases: 2 3 7y x x= − + 3 24 7 21 6y x x x= − + − ( 1)( 3)y x x x= − + • Express the following in index notation and hence find the derivative of each:

1f( ) ( 1)x x xx

= − +

( 1)( 5)f( ) x xxx

− +=

Extend these to include more cases of negative indices for more able students. Give them the chance to practise algebraic techniques, both fractions and indices, according to ability.



2 hours

Differentiation and problem solving


Break down complex problems into smaller tasks.

Understand that the limit

0

f( ) f( )limh

x h xh→

+ − for the

function y = f(x) is represented geometrically as the slope of the tangent to the curve for this function at the point (x, f(x)).

A return to coordinate geometry

Class discussion Remind students of the starting point for this work, the problem of calculating the gradient of the tangent at a point on a curve. Do this now using Autograph, which will generate the derived function’s own graph. Give students time to understand the diagram, and to appreciate the connection between the technique of differentiation and its geometric meaning.

Now pose the question of calculating not just the gradient of a curve as a function but the equation of the tangent to a curve in a specific case. Ask students: • how to find the equation of a line given a point and a gradient; • how to connect that process with the current work.

Then work an example such as finding the tangent at (3, 26) to the curve 3 3y xx

= − . A model

solution is shown on the right.

For more able students, pose a specific question that can be developed to make further connections with past work. For example: • Find the equation of the tangent to the curve y = x3 at the point (2, 8). Show that x3 –12x + 16 = (x – 2)2(x + 4). Find where the tangent at (2, 8) intersects the curve again.

This reinforces calculus, algebra and geometry.

Model solution

3

3 1

2 2

22

3

3d 3 3( 1)d

33

y xx

x xy x xx

xx

−

−

= −

= −

⇒ = − −

= +

At (3, 26) we have x = 3, y = 26 and m (= dy/dx) = 27 1⁄3. So the tangent equation is:

13

13

26 27 ( 3)27 56

y xy x

− = −⇔ = −

Exercises Set questions which relate skills in calculus to coordinate geometry in as many ways as possible. Include: • practice of differentiation; • use of gradients in geometrical applications; • use of algebra to solve geometric problems which result, e.g. simultaneous equations.

Using ICT You could use the online applet Slope from www.fi.uu.nl/wisweb/welcome_en.html.



2 hours

Rates of change and problem solving


Interpret the derivative as a rate of change.

Differentiation and rates of change

Class discussion Remind students of the use of gradients in the context of distance, speed and time graphs: • gradient on a distance–time graph represents velocity; • gradient on a velocity–time graph represent acceleration.

Now discuss a problem that uses these ideas. For example: • The height of a particle t seconds after projection vertically upwards with speed 10 m/s is

given by h = 10t – 5t2. Find when the particle’s speed is: a. 5 m/s; b. zero.

Model solution to the problem

2 d10 5 10 10dhh t t tt

= − ⇒ = −

If the particle’s speed is 5 m/s, then we have: d 5d

5 10 10 0.55 10 10 1.5

ht

t tt t

= ±

= − ⇒ =− = − ⇒ =

If the particle’s speed is zero, then we have:

d 0 1dh tt

= ⇒ =

The particle has speed 5 m/s after 0.5 seconds and after 1.5 seconds, and zero speed after 1 second.

Exercises Set questions which explore this idea. Include: • questions which repeat the example but with different contexts and functions; • questions which explore the same idea but using velocity and acceleration; • questions which bring all of displacement, velocity and acceleration together so that

acceleration arises as a second derivative.


Assessment

Assessment Examples of assessment tasks and questions Notes School resources

The function 2 1f( )

1xxx

−=+

is defined for all x except x = –1, and for these values is the same as the

function g(x) = x – 1. Explain why this is so. For each function find the limit as x tends to –1. What, if any, is the distinction between the functions f and g?

The value of 0

2 2limh

hh→

+ − is

A. 0 B. 12 2

C. 12

D. 12

E. ∞

TIMSS Grade 12

Assessment

Set up activities that allow students to demonstrate what they have learned in this unit. The activities can be provided informally or formally during and at the end of the unit, or for homework. They can be selected from the teaching activities or can be new experiences. Choose tasks and questions from the examples on the right to incorporate in the activities.

Find f ′(x) for f(x) = x2.

How does the mathematics change if f(x) = 3x2?

Find f ′(x) for f(x) = xn.

Use the definition of the derivative applied to this function and the binomial expansion for (x + h)n, remembering that, since h is itself small, h2 and higher powers of h are negligibly small.

The force of gravitational attraction between the Earth of mass M and a satellite of mass m is given by Newton’s law of gravitation as F = GMm / r2, where r is the distance between the centre of the Earth and the satellite and G is the universal constant of gravitation. Find F′(r) and interpret its meaning.

Differentiate 3x6 + 2x + 7.

MEI

In the figure, A and B are points on the curve y = √ x with x-coordinates 2 and 2.1 respectively.

Find the gradient of the chord AB.

Stating the points that you use, find the gradient of another chord which will give a closer approximation to the gradient of the tangent to the curve y = √ x at x = 2.

MEI

Not to scale

Unit 11A.11


Assessment Examples of assessment tasks and questions Notes School resources

The equation of a curve is y = x2 – 8x + 12.

a. Find the points of intersection of this curve with the x- and y-axes.

b. Show that the equation of the tangent to the curve at the point with x-coordinate 5 is y = 2x – 13.

c. Show that this line is also a normal to the curve y = –2x2 + 23.5x – 70 at the point (6, –1).

d. Find the x-coordinates of the points of intersection of the two curves: y = x2 – 8x + 12 y = –2x2 + 23.5x – 70 Give your answers to two decimal places.

MEI

A racing car sets off down a straight section of track towards the first corner. Its speed, v m/s, is modelled for the first four seconds of the motion by

v = t3 – 9t2 + 24t, 0 ≤ t ≤ 4

a. Show that the acceleration, a m/s2, of the car at time t is given by a = k(t – 2)(t – 4), where k is a constant to be determined.

b. Sketch the graph of a against t for 0 ≤ t ≤ 4, stating the coordinates of the points where the graph crosses the a-axis and the t-axis.

For 0 ≤ t ≤ 4, calculate:

c. the greatest speed of the car;

d. the greatest acceleration and the greatest deceleration of the car.

MEI (part)

237 | Qatar mathematics scheme of work | Grade 11 advanced | Unit 11A.12 | Vectors © Education Institute 2005

GRADE 11A: Vectors

Introduction to vector methods

About this unit This is the only unit on vectors for Grade 11 advanced. In it students are introduced to vectors and their elementary properties, including the scalar product.





Previous learning To meet the expectations of this unit, students should already be able to use Pythagoras’ theorem to find the distance between two points defined by Cartesian coordinates.

Expectations By the end of the unit, students will identify and use connections between mathematical topics. They will begin to use vectors to solve physical problems. They will interpret a translation as a vector displacement. They will add and subtract two vectors in up to three dimensions and draw corresponding vector diagrams. They will find the magnitude and direction of a vector, and use vectors to calculate displacement and velocity in a range of contexts.

Students who progress further will develop a facility for connecting mathematical topics, particularly between vectors, geometry and trigonometry. They will use vectors in up to three dimensions and represent them as column matrices. They will know the distinction between a vector and a scalar, and will find scalar products. They will find the magnitude of any vector and its direction in relation to specified axes.

Resources The main resources needed for this unit are: • overhead projector (OHP) • Internet access, computer and data projector • (optional) dynamic geometry system (DGS) such as:


• (optional) computers with Internet access and dynamic geometry software for students


Key vocabulary and technical terms Students should understand, use and spell correctly: • vector, scalar, component, magnitude, direction, sense • vector sum, vector triangle, resultant • section formula, position vector, scalar product, unit vector, basis • commutative, associative • collinear

UNIT 11A.12 6 hours







11A.10.1 Consider coordinate systems as grids for moving around space in two or three dimensions; understand position vector, unit vector and components of a vector.

12AS.13.1 Use vectors in up to three dimensions; identify the components of the vector in relation to three orthogonal directions; use unit vectors i, j and k in these directions; use column matrix form for vectors, including unit vectors; use the notation ABuuur

to denote the vector from point A to point B; use and understand the terms position vector and displacement vector.

11A.10.2 Interpret a translation as a vector displacement; know that a vector displacement from A to B depends only on the starting point A and the finish point B and not on intermediate steps from A to C to D to … to B, and that the vector sum of all these separate displacements from A to B is equivalent to the resultant displacement from A to B directly.

12AS.13.2 Know the rules for the addition and subtraction of two vectors; represent addition and subtraction of two vectors diagrammatically; know that there exists a null vector 0 such that a – a = 0 for any vector; know that vector addition is commutative and associative.

11A.10.3 Add and subtract two vectors in up to three dimensions and draw corresponding vector diagrams.

11A.10.4 Multiply a vector by a scalar and know that this amounts to stretching the vector; calculate the magnitude and direction of a vector; use vectors to calculate displacement and velocity in a range of contexts.

12AS.13.4 Know the distinction between a vector and a scalar; know that any vector can be multiplied by a positive scalar to rescale it, or by a negative scalar to rescale it and reverse its direction.

10A.6.8 Use Pythagoras’ theorem to find the distance between two points.

11A.10.5 Use the scalar product of two vectors to calculate the angle between the vectors and the scalar product of a vector with itself to find the magnitude of the vector.

12AS.13.5 Know the notation a.b for the scalar product of two vectors a and b; form and calculate the scalar product, and interpret the scalar product in terms of the magnitudes of the two vectors and the angle between them; know that a.a is the square of the magnitude of a.

3 hours

Vector algebra 1

1 hour

Vector algebra 2

2 hours

Scalar products

12AS.13.6 Know that if a and b are two non-zero vectors and a.b = 0 then a and b are perpendicular to each other.

11A.10.6 Solve physical problems using vectors.

Unit 11A.12


Activities


3 hours

Vector algebra 1 Consider coordinate systems as grids for moving around space in two or three dimensions; understand position vector, unit vector and components of a vector.

Interpret a translation as a vector displacement; know that a vector displacement from A to B depends only on the starting point A and the finish point B and not on intermediate steps from A to C to D to … to B, and that the vector sum of all these separate displacements from A to B is equivalent to the resultant displacement from A to B directly.

Add and subtract two vectors in up to three dimensions and draw corresponding vector diagrams.

Solve physical problems using vectors.

Calculate the magnitude and direction of a vector; use vectors to calculate displacement and velocity in a range of contexts.

Vectors and translations

Class discussion Draw a coordinate grid and a rectangle with sides parallel to the axes; draw the same rectangle in a different position translated from its original place. Ask students to specify the transformation. Use the answer to introduce the idea of specifying a pair of coordinate displacements which suffice to determine the new position of any point of the original rectangle. Explain that since the same number pair is sufficient to determine the new position of any point, that number pair is called a vector. Introduce the column notation for such a vector. Point out that the same vector can lie in different parts of the plane (corresponding to the same translation but of different points), but it will always have the same length, direction and sense.

Now use the idea of successive translations – not necessarily parallel – to motivate the definition of vector sum. Build this around the idea of putting the free vectors nose to tail to see what their sum is. Show that the order of translations is irrelevant, as borne out by experience. So the commutative law holds for vectors.

Introduce three vectors. Ask students what problems arise from considering three translations: the commutative law means that the order of two in a sum does not matter, but with three the problem is rather which two to add first. Draw a diagram to illustrate the associative law.

Introduce the idea of position vector and the notation associated with it: • use capital letters for points of a diagram, including the origin O; • use small bold italic letters to denote the position vectors of such points, OA=

uuura (when

handwriting vectors, underline the letters).

Stress that a number pair such as (4, 7) always represents a pair of coordinates and hence a

point, whereas a number pair such as 47⎛ ⎞⎜ ⎟⎝ ⎠

always represents a vector. The connection

between the two is that a point always has the column with the same number pair as its position vector.

Generalise these ideas to three dimensions.

Ask students how to find the length of a vector. The answer depends on whether the vector is two- or three-dimensional. Show how Pythagoras’ theorem in two dimensions can be developed into a method for coping with distance in three. (More able students may be willing to generalise this to n dimensions.)

The nose-to-tail arrangement for adding two vectors

The two vectors marked a must have the same length and direction so the quadrilateral is a parallelogram. So the other two sides represent the same vector b. Then the diagonal is either a + b or b + a, so they are equal vectors too. This is the commutative law.

In this diagram, if a and b are added first to get a + b and the result is added to c, we get x = (a + b) + c. On the other hand if we add b and c first, then add a to b + c, we get x = a + (b + c). Thus we have the associative law.


Unit 11A.12



There are several useful tools for vector problems; discuss these with students before starting exercises.

• AB = −uuur

b a for any two points A and B.

• The section formula: if P divides AB in the ratio m : n, then we have ;n mm n

+=+

a bp and this

includes the case of external division where one of m and n is negative. (See right.)

• Three points A, B and C are collinear if and only if AB BCλ=uuur uuur

for some scalar λ, and this

applies for A, B, C in any order.

Exercises Set questions which explore the ideas and the notation.

Examples • If A is the point (4, –5) and B (3, 10) find: – the position vector of A; – the vector which joins A to B; – the length of AB. • If P is the point (7, 9) and Q (–4, 3) find: – p; – PQ;

uuur

– QP;uuur

– p – q; – q – p; – QP; – PQ.

Which of these are equal? Illustrate them in diagrams.

• The forces 432

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

newtons and 572

⎛ ⎞⎜ ⎟−⎜ ⎟−⎝ ⎠

newtons act on a particle. Find the resultant and its

magnitude. • ABCD is a parallelogram. M is the mid-point of AB, and T divides DM in the ratio 2 : 1. If

AD=uuur

u and AB=uuur

v find the vector represented by ATuuur

in terms of u and v. Deduce that A, T and C are collinear, and find AT : TC.

Proof of the section formula

AB

AP AB

( )

AP

( )

( ) ( )

mm nmm n

mm n

m n mm n

m n m mm n

n mm n

= −

=+

= −+

= +

= + −+

+ + −=+

+ + −=+

+=+

uuur

uuur uuur

uuur

b a

b a

p a

a b a

a b a

a a b a

a b

On the web A survey of elementary vectors that extends the work a little further than this unit is at www.physics.uoguelph.ca/tutorials/vectors/ vectors.html.

Some interactive Sketchpad resources are at mathforum.org/~klotz/Vectors/vectors.html.

Physical uses of vectors are discussed in the Physics Classroom at www.physicsclassroom.com/ Class/vectors/vectoc.html.



1 hour

Vector algebra 2 Identify and use interconnections between mathematical topics.

Multiply a vector by a scalar and know that this amounts to stretching the vector.

Scalar multiplication and bases

Class discussion Ask students what meaning they would give to, say, 3x. Use the response to define scalar multiplication. Ask if there are any special cases of interest. Use students’ suggestions to consider the case of multiplication by zero, and by a negative number. Connect this with enlargements.

Stress the difference between a vector and a scalar.

Show that any vector abc

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

can be written:

1 0 00 1 00 0 1

ab a b cc

a b c

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟= + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

= + +i j k

where i, j and k are unit vectors parallel to each of the axes. Explain the notion of a basis of vectors and how this is a way of giving meaning to the idea of dimension.

Reconnect this with geometry to establish the result that k(x + y) = kx + ky, i.e. the distributive law for vectors.

Exercises

Set questions which explore these ideas.

The parallels mean that corresponding angles are equal so OAB is similar to OA′B′. The common ratio of sides then gives:

OA OAk k′ ′= ⇒ =uuuur uuur

a a

and similarly k′ =b b .

A B AB( )kk

′ ′ == −

uuuur uuur

b a

But

A Bk k

′ ′ ′ ′= −= −

uuuurb ab a

so finally

A B ( )k k′ ′ = − = −uuuur

b a b ka

2 hours

Scalar products Identify and use interconnections between mathematical topics.

Use the scalar product of two vectors to calculate the angle between the vectors and the scalar product of a vector with itself to find the magnitude of the vector.

Multiplying vectors

Class discussion Students may already have raised the question of multiplying vectors; if so start from that cue. Otherwise start from the distributive law, where vectors and scalars multiply to give vectors. Show how to define the scalar product: a.b = | a | | b | cos θ

geometrically. Do a few examples to stress the following. • Point out the distinction in notation between a the vector and | a | its magnitude. • The scalar product is commutative, i.e. a.b = b.a. • If θ = 0, the scalar product is just | a | | b |. • If θ = π⁄2 the scalar product is just 0. • If θ is obtuse the scalar product is negative.



Continue to derive the more useful expression for scalar product in components.

In the diagram, let A be (a1, a2, a3) and B (b1, b2, b3). Then we have:

( ) ( ) ( )

1 1

2 2

3 3

2 2 2 21 1 2 2 3 3

2 2 2 2 2 21 1 1 1 2 2 2 2 3 3 3 3

AB

AB

2 2 2

b ab ab a

b a b a b a

b b a a b b a a b b a a

= −−⎛ ⎞

⎜ ⎟= −⎜ ⎟−⎝ ⎠

⇒ = − + − + −

= − + + − + + − +

b auuur

uuur

Now make the connection with trigonometry to bring in the cosine rule: 2 2 2

2 2 2 2 2 21 2 3 1 2 3

AB OA OB 2 OA OB cos

2a a a b b b

θ= + −

= + + + + + − a.b

uuur uuur uuur uuur uuur

Now equate the two expressions for 2

ABuuur

to obtain finally:

1 1 2 2 3 3a b a b a b= + +a.b

Continue to derive the result to use for calculation of the angle between two vectors:

1 1 2 2 3 3

1 1 2 2 3 3

cos

cos

a b a b a ba b a b a b

θ

θ

= = + +

+ +⇒ =

a.b a b

a b

Show this in use with an example.

On the web

Some interactive Sketchpad resources which cover scalar product are available at mathforum.org/~klotz/Vectors/vectors.html.

Exercises Set exercises which: • drill the calculation of angles; • use the scalar product to establish the existence of right angles.

Examples • Use the scalar product to find ∠ABC, where A is (5, 4, 3), B (9, 3, –2), C (4, –1, 2). • Triangle PQR has vertices P (5, 7, –5), Q (4, 7, –3), R (2, 7, –4). Use the scalar product to

show that the triangle is right-angled. Can you do this any other way? • Given that the angle between vectors 2i – j + 3k and i + 3j – pk is 1⁄3 π, find p.

Further investigation Car storm chaser (illuminations.nctm.org/tools/index.aspx)


Assessment


A particle is at the point (6, 2). What is its position vector in terms of the unit vectors i and j in the x- and y-directions respectively? Calculate the length (magnitude) of this vector.

Four vectors a, b, c and d are given by a = 2i – 3j, b = 5j + k, c = 4i – 7k and d = 3i + j. Find a + b, b – c, a – b – c. Draw vector diagrams to represent a + d and a – d. What are the components of these two vectors in the i and j directions?

Find the magnitude of each of the vectors a and d. Calculate the angles between these vectors.

A particle moves with constant velocity from A to B. Its position vector at A is a = i + j and its position vector at B is b = 5i – 7j. Calculate the vector displacement from A to B. If distance is measured in metres, show that the distance from A to B is 4√5 metres.

The particle takes 2 seconds to move from A to B. What is its velocity?

i and j are unit vectors in the east and north directions respectively. A ship has position r = 3i + 4j at 1200 hours. It then moves with constant velocity v = 4i – 5j. The velocity is measured in kilometres per hour. What is the speed of the ship (the magnitude of its velocity)? What is the position of the ship at 1500 hours?


In the figure, OACB is a parallelogram, and M is the mid-point of BC.

Vector OA = auuur

and vector OB .= buuur

a. Find, in terms of a and b,

(i) BC;uuur

(ii) BM;uuur

(iii) OM.uuuur

b. Given that OX is a straight line and that OB = BX, prove that AMX is a straight line.

MEI

A particle of mass m kilograms is moving with constant acceleration a, measured in metres per second per second. The total external force F acting on the particle is measured in newtons, and is the vector sum of the individual forces acting on the particle. The relationship between F and a is given by Newton’s second law of motion and is F = ma.

A particle of mass 2 kg is acted upon by two forces F1 = i – j and F2 = 3j. Find the acceleration of the particle and give its magnitude.

Unit 11A.12



In the figure, ABQCDP represents a tent, held up by vertical poles OP and RQ. The axes Ox and Oy are horizontal at ground level, and Oz is vertically upwards. The coordinates of A, B, C, D, P and Q are as shown in the diagram. Lengths are in metres.

a. Find the length of PQ.

b. Show that the vector n1 = 12i – j + 6k is perpendicular to each of the lines AP and PQ.

c. Vector n2 is –12i – j + 6k. Find the angle between the vectors n1 and n2.

d. A rope ME of length 2 metres is stretched from the mid-point M of PQ to the ground. Given that the rope is perpendicular to PQ, find the coordinates of E.

MEI [part]

245 | Qatar mathematics scheme of work | Grade 11 advanced | Unit 11A.13 | Geometry 2 © Education Institute 2005

GRADE 11A: Geometry 2

Transformations

About this unit This is the second of two units on geometry for Grade 11 advanced. It focuses on the geometry of transformations, including plans and elevations, building on work in Grade 10A and Unit 11A.2.





Previous learning To meet the expectations of this unit, students should already recognise 3-D objects from 2-D representations and be able draw the plan and elevation of a 3-D object from sketches and models. They should be able to identify a single transformation mapping a 2-D shape onto its image, and a line of reflection, centre or angle of rotation, scale factor or centre of enlargement. They should be able to draw, on paper and using ICT, the enlargement of a simple plane figure by a positive fractional scale factor.

Expectations By the end of the unit, students will solve routine and non-routine problems in a range of mathematical and other contexts. They will break down complex problems into smaller tasks. They will draw and use plans and elevations. They will translate, reflect, rotate and enlarge two-dimensional geometric objects.

Students who progress further will solve more complex problems.



• dynamic geometry system (DGS) such as: Geometer’s Sketchpad (see www.keypress.com/sketchpad) Cabri Geometrie (see www.chartwellyorke.com/cabri.html)

• computers with Internet access, graph plotting and dynamic geometry software for students


Key vocabulary and technical terms Students should understand, use and spell correctly: • reflection, rotation, enlargement, dilation, translation, transformation • centre of rotation, scale factor, axis of reflection • commutative, identity • plan, elevation

UNIT 11A.13 7 hours



7 hours SUPPORTING STANDARDS Grade 9 standards



11A.1.1 Solve routine and non-routine problems in a range of mathematical and other contexts, including open-ended and closed problems.



11A.1.9 Generalise whenever possible.

9.5.6 Identify a single transformation mapping a 2-D shape onto its image: reflection, rotation, translation or enlargement by a positive integer scale factor; find a line of reflection, centre or angle of rotation, scale factor or centre of enlargement in simple cases.

11A.9.1 Transform rectilinear figures using combinations of translations, rotations about a centre of rotation, enlargements about a centre of enlargement, and reflections about a line; understand the meanings of positive, negative and fractional scale factors in enlargements.

9.5.7 Identify and draw, on paper and using ICT, the enlargement of a simple plane figure by a positive fractional scale factor; identify the scale factor as the ratio of two corresponding line segments.

11A.9.2 Visualise the effect of transformations on a plane figure; know that the image of a planar figure under rotation or reflection is congruent to the original figure before rotation or reflection and that every circle is similar to any other circle.

2 hours

Harder enlargements

3 hours

Composition of transformations

2 hours

Problem solving with plans and elevations

9.5.9 Recognise 3-D objects from 2-D representations; draw the plan and elevation of a 3-D object from sketches and models; sketch or build a 3-D object given its plan and elevation.

11A.9.3 Draw the plan and elevation of three-dimensional rectilinear objects.

Unit 11A.13


Activities


2 hours

Harder enlargements Understand the meanings of positive, negative and fractional scale factors in enlargements.

Using vectors to describe enlargements

Class discussion Ask students for their ideas on enlargements (also known as dilations). The responses should include some or all of: • angles unchanged; • sides in proportion; • centre and scale factor to be prescribed.

Use these responses to express an enlargement in vector terms: if O is the centre of the enlargement, k is the scale factor and P′ is the image of P, then OP OP.k′ =

uuuur uuur

Ask students if there are any special cases or limitations on the value of k. Special cases would be: • k = 0 (all points map to the origin); • k = 1 (identity transformation); • k = –1 (reflection in the origin).

Explain the term identity if necessary. The possibility of k being negative may be a new idea for some students; move on from k = –1 to consider k negative in general. Show how the enlargement is constructed in this case for (say) a triangle. Ask students in what way the result differs from what they have seen before; they may mention that the image is oriented differently.

Using ICT Use a dynamic geometry system (DGS) to explore this further using the dynamic facility.

[P, –2], say, is a convenient shorthand for the enlargement of centre P, scale factor –2.


Exercises Set exercises to explore this idea.

Examples Do these either on squared paper or using DGS. • A (–1, 1), B (3, 1), C (3, 3) and D (–1, 3) are the vertices of a rectangle. P is the point (2, 0).

Draw ABCD, and its image A1B1C1D1 under the enlargement [P, –3]. Write down the coordinates of A1, B1, C1 and D1.

• List the coordinates of six points on the line y = 2x + 4. List the coordinates of their images under [O, 3], where O is the origin. Find the equation of the line on which the images lie.

Practice using ICT Use Transformations – Dilation to interact with and see the result of a dilation transformation (nlvm.usu.edu/en/nav/vlibrary.html).

Unit 11A.13



Successive transformations

Investigation Use DGS or Autograph as the basis of an investigation into combinations of transformations. Here is an example of how to structure it. • Reflection in two parallel lines: – Draw the points (1, 3), (2, 1), (4, 2) of a triangle T. Draw the line x = –1. – Reflect triangle T in the line x = –1 and call the result U. Reflect U in the line x = 0 and call the result V. How is result V related to triangle T? • Repeat for three other points of your choice, and two other parallel lines. What generalisation

can you make from your conclusions? • Repeat the investigation on triangle T, but this time with successive reflections in x = –1 and y = 2.

• Repeat with a triangle of points of your own choice and two non-parallel lines. What generalisation can you make from your conclusions?

• Repeat the investigations for two successive rotations. What generalisation can you make from your conclusions?

• Repeat the investigations for two successive enlargements about the same centre. What conclusion do you draw?

• Repeat the investigations for two successive rotations about a common centre. What conclusion do you draw?

• Try other combinations of transformations. Be careful to do so systematically.

Two reflections in parallel lines in Autograph

The result is a translation (perpendicular to the lines and twice the distance between them).

Two reflections in non-parallel lines

The result is a rotation about the intersection of the reflection axes and of twice the angle between them.

3 hours

Composition of transformations Solve routine and non-routine problems in a range of mathematical and other contexts, including open-ended and closed problems.

Generalise whenever possible.

Transform rectilinear figures using combinations of translations, rotations about a centre of rotation, enlargements about a centre of enlargement, and reflections about a line.

Visualise the effect of transformations on a plane figure; know that the image of a planar figure under rotation or reflection is congruent to the original figure before rotation or reflection and that every circle is similar to any other circle.

Class discussion From the results of the investigation summarise the conclusions. Students should have found at least the following: • two reflections in parallel lines are equivalent to a translation; • two reflections in non-parallel lines are equivalent to a rotation about the point where the lines

intersect; • two successive enlargements about the same point result in a single enlargement with their

scale factors multiplied together; • two rotations about the same point result in a single rotation of the sum of the two angles.

Ask students to distinguish the transformations which preserve congruence. To give a fully correct answer requires some care about special cases.

Summarise this by checking that students have realised that the image of a planar figure under rotation or reflection is congruent to the original figure before rotation or reflection and that every circle is similar to any other circle.

On the web MathsNet has interactive pages on transformations. Start, for example, at www.mathsnet.net/transform/index.html.

There is also a page there on combining transformations. Its address is www.mathsnet.net/transform/comindex.html.



Exercises Set mixed questions on the various transformations and possible combinations.

Examples • In ABC, ∠BAC = 40°. AB1C is the image of ABC under reflection in the line of AC;

AB1C1 is the image of AB1C under reflection in the line of AB1. – Describe exactly the single transformation which maps ABC to AB1C1. – How many such pairs of reflections will map ABC onto itself? • OA, OB, OC and OD are four lines in that order intersecting at O. Let M1, M2, M3 and M4

represent the operations of reflection in OA, OB, OC and OD respectively. Describe the single operations equivalent to:

– M1M2 (i.e. M2 first, followed by M1); – M3M4; – (M1M2)(M3M4); – (M4M3)(M2M1); – (M3M4)(M1M2). • What single transformation is equivalent to each of the following? – reflection in the x-axis followed by a quarter turn anticlockwise about the origin; – reflection in the y-axis followed by a quarter turn clockwise about the origin; – reflection in y = x followed by a half turn about the origin; – reflection in y = –x followed by a half turn about the origin. • Which of these combinations is commutative? – two reflections; – two rotations; – two translations; – two enlargements.

More practice using ICT Use Transformations – Composition from nlvm.usu.edu/en/nav/vlibrary.html to explore the effect of applying a composition of translation, rotation and reflection transformations to objects.



Harder drawings

Class discussion Put this problem to students. Draw the plan, and front and side elevations, of a pyramid VABCD standing on a base ABCD

6 cm square, and having slant edges 10.5 cm long, if two edges of the base make an angle of 20° with the plan’s baseline.

Ask students how they will start. The response has to be to begin with the base. This must be drawn so that one side is angled at 20° to the plan’s baseline. Use Sketchpad (or other version of DGS) to draw this, with students providing prompts.

Once the square is drawn, ask students for the next step. The response this time must be to work out the height of the pyramid, which is not given directly. Do this by constructing an isosceles triangle with side 10.5 cm on one side of the square.

Show how to begin the elevation constructions by constructing perpendiculars and parallels to the baseline and (arcs of) circles to transfer the measurements. Then finish the construction.

Get students to do all this in DGS as a small project.

Plans and elevations using ICT

The plan and elevation as produced in Sketchpad.

DGS removes much of the work in such a construction, and also makes students think more about what they are doing to complete all the steps.

2 hours

Problem solving with plans and elevations Solve routine and non-routine problems in a range of mathematical and other contexts, including open-ended and closed problems.

Break down complex problems into smaller tasks.

Use a range of strategies to solve problems, including working the problem backwards and then redirecting the logic forwards; set up and solve relevant equations and perform appropriate calculations and manipulations; change the viewpoint or mathematical representation, and introduce numerical, algebraic, graphical, geometrical or statistical reasoning as necessary.

Draw the plan and elevation of three-dimensional rectilinear objects.

Exercises Set further questions of this type that require an essential measurement to be determined indirectly.

Using ICT

Extend the exercises using ICT. For example, use Building houses with side views from www.fi.uu.nl/wisweb/welcome_en.html. Given the side views, build the ‘house’. Rotate the house to see it from different perspectives. There are two levels, with 10 questions at each level.


Assessment


An equilateral triangle ABC has side length 10 cm. It rotates around the inside of a square of side length 20 cm.

a. Triangle ABC rotates about C to the position shown as CA1B1. What is the angle of rotation?

b. Calculate the distance along the path travelled by point A in turning from A to A1.

c. Calculate the distance along the path travelled by point A in turning from A1 to A2.

d. The triangle continues rotating around the inside of the square in the same way until it is back at the original position. Which of the original points A, B or C will point A land on when it has completed its rotations around the inside of the square?


The rectangle Q in the diagram on the right CANNOT be obtained from the rectangle P by means of a:

A. reflection about an axis in the plane of the page

B. rotation in the plane of the page

C. translation

D. translation followed by a reflection

Circle the correct answer.

TIMSS Grade 12

A triangle has vertices at the points (4, 5), (6, 1) and (8, 11). The triangle is enlarged by a factor of 2 about a centre of enlargement at the point (3, –3). Draw the enlarged triangle in its correct position on a coordinate grid.

Line segment OA is 3.0 cm long. Line segment OB is √ 7 cm long. OB can rotate in a horizontal plane about the point O.

Find the maximum possible distance B can be from A. Explain whether your answer is a rational number or an irrational number.

Find the minimum possible distance B can be from A. Explain whether your answer is a rational number or an irrational number.

Sketch a different position for line segment OB so that the distance from A to B, AB, is a rational number. Confirm by calculation that your answer is a rational number.

OB is reduced in length to 2.6 cm. OA is still 3.0 cm long. Calculate the distance AB when angle AOB is 120°.

The lengths of 2.6 cm and 3.0 cm are accurate to one decimal place. The 120° angle is accurate to the nearest degree. Calculate the greatest and least possible values of AB.

Unit 11A.13



The diagram shows parts of two circles, sector A and sector B.

a. Which sector has the bigger area?

b. The perimeter of a sector is made from two straight lines and an arc. Which sector has the bigger perimeter?

A stone monument consists of a square pedestal of edge 3 metres and height 1 metre topped by a cone of radius 1.5 metres and height 3 metres so that its base just fits the top of the pedestal. Use DGS to draw the plan and elevation of the figure with respect to a baseline which makes an angle of 30° with an edge of the base.

A semicircle, of radius 4 cm, has the same area as a complete circle of radius r cm. What is the radius of the complete circle?



Transformations of graphs, exponentials and logarithms

About this unit This is the last of four units on algebra for Grade 11 advanced. In it students learn more about functions and their transformations, and about exponentials and logarithms.





Previous learning To meet the expectations of this unit, students should understand and be able to apply the laws of indices and to set up functional relationships between related variables.

Expectations By the end of the unit, students will use mathematics to model and predict outcomes of real-world applications, and will compare and contrast two or more given models of a particular situation. They will identify and use connections between mathematical topics. They will solve problems using inverse and composite functions. They will apply combinations of transformations to the graph of the function y = f(x). They will understand exponential growth and decay and plot the graphs and the natural logarithm function using a graphics calculator, and will use the log function (logarithm in base 10) on a calculator. Through their study of functions and their graphs, and the solution of associated equations, students will appreciate a range of numerical and algebraic applications in the real world.

Students who progress further will recognise that the exponential and logarithm functions are key to solving a wider range of problems.




Key vocabulary and technical terms Students should understand, use and spell correctly: • exponential, logarithm • transformation, stretch (shear), translation, rotation, reflection,

enlargement • inverse, composite, construct, deconstruct










10A.3.1 Understand exponents and nth roots, and apply the laws of indices to simplify expressions involving exponents; use the xy key on a calculator.

11A.3.1 Develop further confidence in using the rules for indices. 12AS.3.1 Understand exponents and nth roots, and apply the laws of indices to simplify expressions involving exponents; use the xy key and its inverse on a calculator.

10A.5.2 Understand and use the concept of related variables and, in special cases, set up appropriate functional relationships between them.

11A.5.17 Understand the transformations of the function y = f(x) given by: • y = f(x) + a, representing a translation by a in the

positive y-direction; • y = f(x + a), representing a translation by –a in the

positive x-direction; • y = a f(x), representing a stretch with scale factor a

parallel to the y-axis; • y = f(ax), representing a stretch with scale factor 1 / a

parallel to the x-axis.

Use these and combinations of these transformations to sketch, stage by stage, the transformation of the graph of y = f(x) into the graph of the transformed function.

12AS.5.4 Form composite functions and use the notation y = g(f(x)).

12AS.5.1 Use a graphics calculator to plot exponential functions of the form y = ekx; describe these functions, distinguishing between cases when k is positive or negative, and the special case when k is zero.

5 hours Transformations of graphs

6 hours

Logarithms and exponential functions

11A.5.18 Understand the ideas of exponential growth and decay and the forms of the associated graphs y = ax, where a > 0; use a graphics calculator to plot the graphs of the exponential function, ex, and the natural logarithm function, ln x; know that one is the inverse function of the other and use this to find solutions to physical problems; solve for x the equation y = ax and use this in problems; use the log function (logarithm in base 10) on a calculator.

12AS.5.2 Plot and describe the features of the natural logarithm function y = ln x; understand that the natural logarithm function is inverse to the exponential function.

Unit 11A.14


Activities


Transformations

Class discussion Ask students what they remember about transformations. Use the discussion to review: • translation; • reflection; • rotation; • stretching (shear); • enlargement.

Except for rotation and enlargement, they will study these again in what follows.


5 hours Transformations of graphs


Understand the transformations of the function y = f(x) given by: • y = f(x) + a, representing a

translation by a in the positive y-direction;

• y = f(x + a), representing a translation by –a in the positive x-direction;

• y = a f(x), representing a stretch with scale factor a parallel to the y-axis;

• y = f(ax), representing a stretch with scale factor 1 / a parallel to the x-axis.

Use these and combinations of these transformations to sketch, stage by stage, the transformation of the graph of y = f(x) into the graph of the transformed function.

Investigation Give students a range of functions to graph using graph plotting software (better) or graphics calculators (acceptable). For examples, see the notes on the right.

For trigonometry examples, use radian measure (this is to avoid large numbers).

Ask students to investigate systematically what happens geometrically in the case of each of the transformations on the right. Do an example to illustrate how to effect the transformation on the calculator or graph plotting software. Give students advice on how to record the work as it proceeds. This is to avoid them being so overwhelmed by the quantity that they are unable to see resulting patterns. If using graph plotting software, save the files; with a calculator, make indexed sketches.

Give this the time it needs. Encourage discussion before summarising the conclusions.

Examples for students to graph • y = 3x • y = –2x + 1 • y = x2 • y = 5 – 4x – x2 • y = 2x3 – 5x2 + 4x – 7 • y = sin x • y = tan (2x – 3)

Transformations to investigate • Change x to (x + 3). • Change x to (x – 2). • Change f(x) to f(x) + 3. • Change f(x) to f(x) – 2. • Change x to (3x). • Change x to (–2x).

• Change x to (1⁄2 x). • Change f(x) to 3 f(x). • Change f(x) to –2 f(x). • Change f(x) to 1⁄3f(x).

Unit 11A.14



Class discussion

Students met the idea of composition (and decomposition) in Unit 11A.8 (Algebra 3). The aim now is to connect this idea with the transformations just considered. Discuss an example and spell out the steps of the decomposition in parallel with the geometric development.

Example y = 2 – 4 sin (5x + 3)

x → sin x start with the standard sine wave (using radian measure)

x → sin (x + 3) translation 30

−⎛ ⎞⎜ ⎟⎝ ⎠

x → sin( 5x + 3) stretch parallel to Ox with scale factor 1⁄5

x → –4 sin (5x + 3) stretch parallel to Oy with scale factor –4

x → 2 – 4 sin (5x + 3) translation 02⎛ ⎞⎜ ⎟⎝ ⎠

The diagrams on the right show the various stages of the process. Tell students that they should be able to sketch the steps of this by careful argument and knowledge of the transformations.

y = sin x

y = sin (x + 3)

y = sin (5x + 3)

y = –4 sin (5x + 3)

y = 2 – 4 sin (5x + 3)

Exercises Set questions which require this technique. Begin with explicit equations, so that the result (final or step by step) can be checked on a graphics calculator; increase the difficulty by providing sketches of unspecified functions so that students must think geometrically. To be successful, students must be able to do this without recourse to squared paper.

Example on an unspecified function Provide a sketch of an unknown function y = f(x) and use it to sketch: • y = f(2x); • y = 3f(x + 3).



6 hours

Logarithms and exponential functions

Use mathematics to model and predict the outcomes of real-world applications; compare and contrast two or more given models of a particular situation.

Develop further confidence in using the rules for indices.

Understand the ideas of exponential growth and decay and the forms of the associated graphs y = ax, where a > 0; use a graphics calculator to plot the graphs of the exponential function, ex, and the natural logarithm function, ln x; know that one is the inverse function of the other and use this to find solutions to physical problems; solve for x the equation y = ax and use this in problems; use the log function (logarithm in base 10) on a calculator.

Graphs and their gradients

Class discussion Students have met gradients of curves in three earlier units of Grade 11; this section will build on that knowledge.

Ask students to sketch the graph of y = ax. Once they have become interested in the problem, draw the graph using Autograph or other graph plotting software.

Autograph will assign a default value of a = 1. Vary the constant value in Autograph (View → Constant Controller) so that students can see the variation in shape as the value of a increases or decreases. On the same axes show the gradient of the graph. Ask students to explain how to interpret the picture.

Now pose the question of how to get the two curves to coincide. It is easy to get the value 2.7 to achieve this. Tell students that a more precise value is 2.7183, and, if appropriate, mention also

that the number can be shown to be 1 11 1 ... e.2! 3!

+ + + + =

Show also that with the same value e the function y = Aex has the same property (i.e. curve and gradient curve coincide).

Explain that y = Aex, where A is a constant, is the solution to the problem of finding a function

which satisfies d .d

y yx

=

Ask the class why this may be useful, and introduce the idea of growth and decay. For example: • animal populations are expected to multiply in proportion to the number already in existence; • radioactive substances produce emissions and decay in proportion to the current quantity.

Using transformations, as on the right, show that y = Aekx is the solution of d .d

y kyx

= Show how

to use this to solve simple cases of the differential equation of growth or decay.

Autograph display showing the use of the

constant controller to find an approximation for e

Stretching the graphs above with factor k–1 parallel to the x-axis in the case that they coincide (a = e) transforms y = ex into y = ekx and dd

y yx

= into dd

y kyx

= since the y-values are

unchanged in the stretch but the curve is k times steeper. This is a subtle argument but it eliminates the need for the chain rule.

Exercises Set questions which explore the use of the new function, including: • sketch graphs, using the graphics calculator; • scientific examples of growth and decay; • simple applications of the differential equation and its solution; • development of different models for comparison, e.g. a growth function with different

proposed parameters compared with linear variation – this is an opportunity for cross-curricular work.

On the web

MathsNet has pages on this and later sections: www.mathsnet.net/asa2/2004/c3.html#3.

There is also a page devoted to the properties of this and other exponential functions at Visual Calculus: archives.math.utk.edu/visual.calculus/ 0/exp_log.5/.



From logarithms to exponentials

Class discussion Remind students of the relation between the graph of a function and that of its inverse: one is a reflection of the other in the line y = x. Use this idea to introduce the idea of the logarithm function; it is the inverse of y = ex. Focus on this by using the notation: y = ex

⇔ x = ln y

Point out that values of this function are available on calculators.

Now develop the properties of this new function, using this notation to link it directly to the exponential function. a = ln p and b = ln q ⇒ p = ea and q = eb ⇒ pq = ea eb = ea+b ⇒ ln (pq) = a + b

= ln p + ln q

Derive expressions for ln pq

⎛ ⎞⎜ ⎟⎝ ⎠

similarly. Use the special case of q = 1 to show that ln 1 = 0.

Ask students to simplify ln (xn) and ln (x–n), where n is a positive integer. Check that students begin by changing them to exponential functions. Discuss this, and conclude that: ln (xn) = n ln x and ln (x–n) = –n ln x

Invite students to conjecture how ln (xm/n) simplifies. When the correct answer is in view, allow sufficient discussion to show that this does not follow just by changing n to m/n in the previous result(because m/n is not a positive integer). Then demonstrate how to use the previous result to show that the equivalent result is nonetheless valid:

( )/

/

ln( )

ln ( )

ln( )ln

ln

m n

m n n

m

p x

np x

xm x

mp xn

=

⇒ =

==

⎛ ⎞⇒ = ⎜ ⎟⎝ ⎠

Graphs of the exponential function and its inverse

Exercises Set questions which apply simple properties of logarithms, such as: • Express ln 6, ln 12 and ln 1.5 in terms of ln 2 and ln 3.



Logarithms to any base

Class discussion Ask students for what values of x they can give a precise value for ln x (e.g. x = 1, e and other values such as e2). Introduce the problem of calculating (say) log3

2. To do this, express the problem in index terms and then take logarithms to a base available on a calculator (base e or 10). Thus

3

3

log 2

2 3ln2 ln3

ln2ln3ln2i.e. log 2ln3

p

p

p

p

=

⇒ =⇒ =

⇒ =

=

This is a special case of the general result logloglog

cb

c

xxb

= which is proved similarly and is a

good challenge for more able students.

With students, convert a value of their choice from base e to base 10.

The theory studied here does not answer the question of how the logarithms given on a calculator are worked out. Encourage students who persist with that question to consider the series

2 3

ln(1 ) ... ( 1 1)2! 3!x xx x x+ = − + − − < ≤

as a starting point.

Exercises Set questions which cover all these ideas.

Examples • Calculate log4

5. • Solve the equation 3x = 4. • Solve the equation ln x + ln (x + 3) = ln 4. • Solve the equation 4x – 3 × 2x + 2 = 0.

• Sketch on the same axes the graphs of y = ln x, y = ln (x + k) and y = ln (kx), making clear how they are related.

• Find the image of the curve y = ex:

– after a translation parallel to the x-axis of ;0k−⎛ ⎞

⎜ ⎟⎝ ⎠

– after an additional stretch parallel to the y-axis of scale factor b. What happens when b = e–k?

On the web An article which looks ahead from this point is at www.hyper-ad.com/tutoring/math/calculus/ The%20Natural%20Logarithm%20Function.html.


Assessment


Curve B is the translation, one unit up the y-axis, of y = x2. What is the equation of curve B?

Translate curve B two units to the left. What is the equation of this new curve?

Describe in words how the graph of y = 1 / x is transformed into the graph y = 4 + 5 / x. Sketch each graph on the same set of axes.

The growth of the Internet since 1990 has been modelled by the function N = 0.2(1.8)t, where N is the number of users, counted in millions, t years from 1990. Plot the graph of this function. How many Internet users does the model predict for the year 2006?

Assessment Set up activities that allow students to demonstrate what they have learned in this unit. The activities can be provided informally or formally during and at the end of the unit, or for homework. They can be selected from the teaching activities or can be new experiences. Choose tasks and questions from the examples on the right to incorporate in the activities. When living organisms die the amount of carbon-14 present in the dead matter decays exponentially

according to the formula N = N0e–0.000121t, where N0 is the initial quantity and t is the time in years. A bone uncovered at an archaeological site has 35% of its original carbon-14. Estimate the age of the bone. After how many more years will the bone have only 25% of its carbon-14?

The Global Report estimated the population of the world in 1975 as 4.1 billion people and that it was growing at the rate of 2% per year. Set up an equation to predict the world population t years from 1975. Use this model to predict the world’s population in 2020. Discuss any assumptions you make.

Earthquakes produce oscillations in the ground. The strength, S, of the quake is measured on the Richter scale and is given by S = log A, where A is the measured amplitude of the oscillation as measured in millimetres on a calibrated seismograph. What amplitude of oscillation corresponds to a major earthquake with a Richter scale value of 7.8? What is the Richter scale value of an earthquake with an oscillation that has an amplitude of 2000 mm?

The figure shows a sketch of the graph y = f(x) for 0 ≤ x ≤ 4 where f(x) = √(4 – x). Find the inverse function f–1(x). Copy the figure and draw the graph of y = f–1(x) on the same diagram. What is the connection between the graph of y = f(x) and the graph of y = f–1(x)?

The figures below show the graph of y = f(x) together with the graphs of y = f1(x), y = f2(x) and y = f3(x), respectively, each of which is a simple transformation of the graph of y = f(x). Find expressions in terms of x for each of the functions f1(x), f2(x) and f3(x).

The function g(x) is defined in such a way that the composite function gf(x) equals x – 4. Find the functions g(x) and fg(x).

MEI [modified]

Unit 11A.14

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